Evidently, the h-plane contains at least two points. Therefore, to show that Axiom I holds in the h-plane, we need to show that the h-distance defined in 12.1 is a metric on h-plane; that is, the conditions (a) - (d) in Definition 1.3.1 hold for h-distance.

The following claim says that the h-distance meets the conditions (a) and (b)

Claim (PageIndex{1})

Given the h-points (P) and (Q), we have (PQ_h ge 0) and (PQ_h=0) if and only if (P=Q).

**Proof**According to Lemma 12.3.1 and the main observation (Theorem 12.3.1), we may assume that (Q) is the center of the absolute. In this case

(delta(Q,P)=dfrac{1+QP}{1-QP}ge 1)

and therefore

(QP_h=ln[delta(Q,P)] ge 0.)

Moreover, the equalities holds if and only if (P=Q).

The following claim says that the h-distance meets the condition

Claim (PageIndex{2})

For any h-points (P) and (Q), we have (PQ_h=QP_h).

**Proof**Let (A) and (B) be ideal points of ((PQ)_h) and (A,P,Q,B) appear on the circline containing ((PQ)_h) in the same order.

Then

(egin{array} {rcl} {PQ_h} & = & {ln dfrac{AQ cdot BP}{QB cdot PA} =} {} & = & {=ln dfrac{BP cdot AQ}{PA cdot QB}=} {} & = & {QP_h} end{array})

The following claim shows, in particular, that the triangle inequality (which is Definition 1.3.1d) holds for (h)-distance.

Claim (PageIndex{3})

Given a triple of h-points (P), (Q), and (R), we have

(PQ_h+QR_h ge PR_h.)

Moreover, the equality holds if and only if (P), (Q), and (R) lie on one h-line in the same order.

**Proof**Without loss of generality, we may assume that (P) is the center of the absolute and (PQ_h ge QR_h >0).

Suppose that (Delta) denotes the h-circle with the center (Q) and h-radius ( ho=QR_h). Let (S) and (T) be the points of intersection of ((PQ)) and (Delta).

By Lemma 12.3.3, (PQ_hzge QR_h). Therefore, we can assume that the points (P), (S), (Q), and (T) appear on the h-line in the same order.

According to Lemma Lemma 12.3.4, (Delta) is a Euclidean circle; suppose that (hat Q) denotes its Euclidean center. Note that (hat Q) is the Euclidean midpoint of ([ST]).

By the Euclidean triangle inequality

[PT = Phat{Q}+hat{Q} R ge PR]

and the equality holds if and only if (T=R).

By Lemma Lemma 12.3.2,

(egin{array} {l} {PT_h = ln dfrac{1 + PT}{1 - PT},} {PR_h = ln dfrac{1 + PR}{1 - PR}.} end{array})

Since the function (f(x)=lnfrac{1+x}{1-x}) is increasing for (xin[0,1)), inequality 12.4.1 implies

(PT_hge PR_h)

and the equality holds if and only if (T=R).

Finally, applying Lemma 12.3.3 again, we get that

(PT_h=PQ_h+QR_h.)

Hence the claim follows.

Question 1.

Determine whether * is a binary operation on the sets given below

(i) a * b = a.|b| on R ,

(ii) a * b = min (a, b) on A = <1, 2, 3, 4, 5>

(iii) (a * b) = (a sqrt**) is binary on R. Solution: (i) Yes. Reason: a, b ∈ R. So, |b| ∈ R when b ∈ R Now multiplication is binary on R So a|b| ∈ R when a,be R. (Le.) a * b ∈ R. * is a binary operation on R.**

** **

**(ii) Yes. Reason: a, b ∈ R and minimum of (a, b) is either a or b but a, b ∈ R. So, min (a, b) ∈ R. (Le.) a * b ∈ R. * is a binary operation on R.**

(iii) a* b = (a sqrt**) where a, b ∈ R. No. * is not a binary operation on R. Reason: a, b ∈ R. ⇒ b can be -ve number also and square root of a negative number is not real. So (sqrt**

**) ∉ R even when b ∈ R. So (sqrt**

**) ∉ R. ie., a * b ∉ R. * is not a binary operation on R.**** **

Question 2.

Solution:

No. * is not a binary operation on Z.

Reason: Since m, n ∈ Z.

So, m, n can be negative also.

Now, if n is negative (Le.) say n = -k where k is +ve.

Similarly, when m is negative then n m ∉ Z.

∴ m * n ∉ Z. ⇒ * is not a binary operation on Z.

Question 3.

Let * be defined on R by (a * b) = a + b + ab – 1 .Is * binary on R ? If so, find (3 *left(frac<-7><15>
ight))

Solution:

a * b = a + b + ab – 7.

Now when a, b ∈ R, then ab ∈ R also a + b ∈ R.

So, a + b + ab ∈ R.

We know – 7 ∈ R.

So, a + b + ab – 7 ∈ R.

(ie.) a * b ∈ R.

So, * is a binary operation on R.

Question 4.

Let A= )b: a, b ∈ Z>. Check whether the usual multiplication is a binary operation on A.

Solution:

Let A = a + (sqrt<5>) b and B = c + (sqrt<5>)d, where a, b, c, d ∈ M.

Now A * B =)b)(c + (sqrt<5>)d)

= ac + (sqrt<5>)ad + (sqrt<5>)bc + (sqrt<5>)b(sqrt<5>)d

= (ac + 5bd) + (sqrt<5>)(ad+ bc) ∈ A

Where a, b, c, d ∈ Z

So * is a binary operation.

Question 5.

(i) Define an operation * on Q as follows: a * b = (left(frac<2>
ight)) a, b ∈ Q. Examine the closure,

commutative, and associative properties satisfied by * on Q.

(ii) Define an operation * on Q as follows: a*b = (left(frac<2>
ight)) a, b ∈ Q. Examine the existence of identity and the existence of inverse for the operation * on Q.

Solution:

(i) 1. Closure property:

Let a,b ∈ Q.

So, closure property is satisfied.

2. Commutative property:

Let a, b ∈ Q.

(1) = (2) ⇒ Now a * b = b * a

⇒ Commutative property is satisfied.

3. Associative property:

Let a,b,c G Q. ^

To prove associative property we have to prove that a * (b * c) = (a * b) * c

LHS: a * (b * c)

(i.e.) the identity Clement e = a which is not possible.

So, the identity element does not exist and so inverse does not exist.

Question 6.

Fill In the following table so that the binary operation * on A = is commutative.

Solution:

Given that the binary operation * is Commutative.

To find a * b :

a * b = b * a (∵ * is a Commutative)

Here b * a = c. So a * b = c

To find a *c:

a * c = c * a (∵ * is a Commutative)

c * a = a. (Given)

So a * c = a

To find c * b:

c * b = b * c

Here b * c = a.

So c * b = a

Question 7.

Consider the binary operation * defined on the set A = [a, b, c, d] by the following table:

– Is it commutative and associative?

Solution:

From the table

b * c = b

c * b = d

So, the binary operation is not commutative.

To check whether the given operation is associative.

Let a, b, c ∈ A.

To prove the associative property we have to prove that a * (b * c) = (a * b) * c

From the table,

LHS: b * c = b

So, a * (b * c) = a * b = c ……. (1)

RHS: a * b = c

So, (a * b) * c = c * c = a …… (2)

(1) ≠ (2). So, a * (b * c) ≠ (a * b) * c

∴ The binary operation is not associative.

Question 8.

Solution:

Question 9.

and Let * be the matrix multiplication. Determine whether M is closed under * . If so, examine the commutative and associative properties satisfied by * on M .

and let * be the matrix multiplication. Determine whether M is closed under *. If so, examine the existence of identity, existence of inverse properties for the operation * on M.

Solution:

So, inverse property is satisfied.

Question 10.

(i) Let A be Q <1). Define * on A by x * y = x + y – xy. Is * binary on A ? If so, examine the commutative and associative properties satisfied by * on A.

(ii) Let A be Q<1>. Define *on A by x * y = x + y – xy. Is * binary on A ? If so, examine the existence of identity, existence of inverse properties for the operation * on A.

Solution:

(i) Let a,b ∈ A (i.e.) a ≠ ±1 , b ≠ 1

Now a * b = a + b – ab

If a + b – ab = 1 ⇒ a + b – ab – 1 = 0

(i.e.) a(1 – b) – 1(1 – b) = 0

(a – 1)(1 – b) = 0 ⇒ a = 1, b = 1

But a ≠ 1 , b ≠ 1

So (a – 1) (1 – 6) ≠ 1

(i.e.) a * b ∈ A. So * is a binary on A.

To verify the commutative property:

Let a, b ∈ A (i.e.) a ≠ 1 , b ≠ 1

Now a * b = a + b – ab

and b * a = b + a – ba

So a * b = b * a ⇒ * is commutative on A.

To verify the associative property:

Let a, b, c ∈ A (i.e.) a, b, c ≠ 1

To prove the associative property we have to prove that

a * (b * c) = (a * b) * c

LHS: b * c = b + c – bc = D(say)

So a * (b * c) = a * D = a + D – aD

= a + (b + c – bc) – a(b + c – bc)

= a + b + c – bc – ab – ac + abc

= a + b + c – ab – bc – ac + abc …… (1)

RHS: (a * b) = a + b – ab = K(say)

So (a * b) * c = K * c = K + c – Kc

= (a + b – ab) + c – (a + b – ab) c

= a + b – ab + c – ac – bc + abc

= a + b + c – ab – bc – ac + abc ….. (2)

(ii) To verify the identity property:

Let a ∈ A (a ≠ 1)

If possible let e ∈ A such that

a * e = e * a = a

To find e:

a * e = a

(i.e.) a + e – ae = a

So, e = (≠ 1) ∈ A

(i.e.) Identity property is verified.

To verify the inverse property:

Let a ∈ A (i.e. a ≠ 1)

If possible let a’ ∈ A such that

To find a’:

a * a’ = e

(i.e.) a + a’ – aa’ = 0

⇒ a'(1 – a) = – a

⇒ For every a ∈ A there is an inverse a’ ∈ A such that

a* a’ = a’ * a = e

⇒ Inverse property is verified.

### Samacheer Kalvi 12th Maths Solutions Chapter 12 Discrete Mathematics Ex 12.1 Additional Problems

Question 1.

Show that the set G = )/ a, b ∈ Q> is an infinite abelian group with respect to Binary operation addition. Satisfies closure, associative, identity and inverse properties.

Solution:

(i) Closure axiom :

Let x, y ∈ G. Then x = a + b(sqrt<2>), y = c + d(sqrt<2>) a, b, c, d ∈ Q.

x + y = (a + b(sqrt<2>)) + (c + d(sqrt<2>)) = (a + c) + (b + d) (sqrt<2>) ∈ G,

Since (a + c) and (b + d) are rational numbers.

∴ G is closed with respect to addition.

(ii) Associative axiom : Since the elements of G are all real numbers, addition is associative.

(iii) Identity axiom : There exists 0 = 0 + 0 (sqrt<2>) ∈ G

such that for all x = a + b(sqrt<2>) ∈ G.

x + 0 = (a + b(sqrt<2>) ) + (0 + 0(sqrt<2>))

= a + b(sqrt<2>) = x

Similarly, we have 0 + x = x. ∴ 0 is the identity element of G and satisfies the identity axiom.

(iv) Inverse axiom: For each x = a + b(sqrt<2>) ∈ G,

there exists -x = (-a) + (-b) (sqrt<2>) ∈ G

such that x + (-x) = (a + b(sqrt<2>)) + ((-a) + (- b)(sqrt<2>))

= (a + (-a)) + (b + (-b)) (sqrt<2>) = 0

Similarly, we have (- x) + x = 0 .

∴ (- a) + (-b)(sqrt<2>) is the inverse of a + b (sqrt<2>) and satisfies the inverse axiom.

(v) Commutative axiom:

x + y = (a + c) + (b + d) (sqrt<2>) = (c + a) + (d + b) (sqrt<2>)

= (c + d(sqrt<2>)) + (a + b(sqrt<2>))

= y + x, for all x, y ∈ G.

∴ The commutative property is true.

∴ (G, +) is an abelian group. Since G is infinite, we see that (G, +) is an infinite abelian group.

Question 2.

Show that (Z_{7} – < [0]>, .7) write to the binary operation multiplication modul07 satisfies closure, associative, identity and inverse properties.

Solution:

Let G = [[1], [2],… [6]]

The Cayley’s table is

From the table:

(i) all the elements of the composition table are the elements of G.

∴ The closure axiom is true.

(ii) multiplication modulo 7 is always associative.

(iii) the identity element is [1] ∈ G and satisfies the identity axiom.

(iv) the inverse of [1] is [1] [2] is [4] [3] is [5] [4] is [2] [5] is [3] and [6] is [6] and it satisfies the inverse axiom.

Question 3.

Show that the set G of all positive rationals with respect to composition * defined by ab

a* b = (frac<3>) for all a, b ∈ G satisfies closure, associative, identity and inverse properties.

Solution:

Let G = Set of all positive rational number and * is defined by,

(i) Closure axiom: Let a, b ∈ G

∴ closure axiom is satisfied.

(ii) Associative axiom: Let a, b, c ∈ G.

To prove the associative property, we have to prove that

(1) = (2) ⇒ LHS = RHS i.e., associative axiom is satisfied.

(iii) Identity axiom: Let a ∈ G.

Let e be the identity element.

By the definition, a * e = a

e = 3 ∈ G ⇒ identity axiom is satisfied.

(iv) Inverse axiom: Let a ∈ G and a’ be the inverse of a * a’ = e = 3.

Question 4.

Show that the set G of all rational numbers except – 1 satisfies closure, associative, identity and inverse property with respect to the operation * given by a * b = a + b + ab for all a, b ∈ G

Solution:

G = [Q, -<-l>]

* is defined by a * b = a + b + ab

To prove G is an abelian group.

G_{1}: Closure axiom: Let a, b ∈ G.

i.e., a and b are rational numbers and a ≠ -1, b ≠ -1.

So, a * b = a + b + ab

If a + b + ab = – 1

⇒ a + b + ab + 1 = 0

i.e., (a + ab) + (b + 1) = 0

a (1 + b) + (b + 1) = 0

i.e., (a + 1)(1 + b) = 0

⇒ a = -1, b = -1

But a ≠ -1, b ≠ -1

⇒ a + b + ab ≠ -1

i.e., a + b + ab ∈ G ∀ a, b ∈ G

⇒ Closure axiom is verified.

G_{2}: Associative axiom: Let a, b, c ∈ G.

To prove G_{2}, we have to prove that,

a *

LHS:

b * c = b + c + bc = D (say)

a * (b * c) = a * D = a + D + aD

= a + (b + c + bc) + a(b + c + bc)

= a + b + c + bc + ab + ac + abc

= a + b + c + ab + bc + ac + abc ……. (1)

RHS:

a * b = a + b + ab = E (say)

∴ (a * b) * c = E * c = E + c + Ec

= a + b + ab + c + (a + b + ab) c

= a + b + ab + c + ac + bc + abc ……. (2)

= a + b + c + ab +be+ ac + abc

(1) = (2) ⇒ Associative axiom is verified.

G_{3}: Identity axiom: Let a ∈ G. To prove G_{3} we have to prove that there exists an element e ∈ G such that a * e = e * a = a.

To find e: a * e = a

i.e., a + e + ae = a

⇒ e(1 + a) = a – a = 0

So, e = 0 ∈ G ⇒ Identity axiom is verified.

G_{4} : Inverse axiom: Let a ∈ G. To prove G_{4}, we have to prove that there exists an element a’ ∈ G such that a * a’ = a’ * a = e.

To find a’: a * a’ = e

i.e., a + a’ + aa’=: 0 <∵ e = 0>

⇒ a'(1 + a) = -a

Thus, inverse axiom is verified.

Question 5.

Show that the set < [1], [3], [4], [5], [9]>under multiplication modulo 11 satisfies closure, associative, identity and inverse properties.

Solution:

G = <[1], [3], [4], [5], [9]>

* is defined by multiplication modulo 11.

To prove G is an abelian group with respect to *

Since we are given a finite number of elements i.e., since the given set is finite, we can frame the multiplication table called Cayley’s table.

The Cayle’s table is as follows:

G_{1}: The elements in the above table are [1], [3], [4], [5] and [9] which are elements of G.

∴ closure axiom is verified.

G_{2}: Consider [3], [4], [5] which are elements of G.

<[3] * [4]>* [5] = [1] * [5] = [5] ……. (1)

[3] * <[4] * [5]>= [3] * [9] = [5] …… (2)

(1) = (2) ⇒ (a * b) * c = a * (b * c) i.e., associative axiom is verified.

G_{3}: The first row elements are the same as that of the given elements in the same order. ie., from the table, the identity element is [1] ∈ G. So identity axiom is verified.

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## 12.4: Axiom I

Zechariah 12:4-5 . In that day — This expression, in the prophetical writings, is of large extent, and not only signifies that particular point of time last spoken of, but some time afterward. I will smite every horse with astonishment — Many commentators explain this of the victories which Judas Maccabæus gained over Antiochus’s captains, whose chief force consisted in cavalry. But, as Archbishop Newcome observes, the language is much too strong, as it is also Zechariah 12:6-9, to denote the successes of the Maccabees against the Seleucidæ. This prophecy therefore, he thinks, remains to be accomplished. And many commentators, who are of the same opinion, consider it as a prediction of victories that will be obtained over Gog and Magog by the Jews, upon their restoration to their own land. One circumstance in favour of this interpretation is, that Gog and Magog are represented, Ezekiel 38:15, as riders on horses. And if by that people the Turks be intended, we know that they have been, and still are, famous for their cavalry, wherein chiefly the strength of their armies consists. But it is here foretold, that in order to their discomfiture God will send such distraction among their horses and their riders, and throw them into such a state of confusion, that they shall fall foul one upon another,

(see Zechariah 14:13,) and not be able to distinguish between their friends and their foes. And I will turn mine eyes upon the house of Judah — I will have an especial concern for their preservation. And the governors of Judah shall say in their heart — Shall say within themselves, The inhabitants of Jerusalem shall be my strength in the Lord — “The text here,” says Blayney,” has been supposed corrupt, and many attempts have been made to amend it. But, without any alteration, it well expresses the sentiments of the men of Judah, concerning the interest they had in the safety of Jerusalem and its inhabitants, on which their own strength and security depended in a great degree so that they would, of course, be influenced to bring that assistance, the efficacy of which is set forth in the verse that follows.”

With one considerable exception , those who would sever the six last chapters from Zechariah, are now at one in placing them before the captivity. Yet, Zechariah here too speaks of the captivity as past. Adopting the imagery of Isaiah, who foretells the delivery from the captivity as an opening of a prison, he says, in the name of God, "By the blood of thy covenant I have sent forth thy prisoners out of the pit wherein is no water" Zechariah 9:11. Again, "The Lord of hosts hath visited His flock, the house of Judah. I will have mercy upon them (Judah and Joseph) and they shall be as though I had not cast them off" Zechariah 10:3-5. The mention of the mourning of all the "families that remain" Zechariah 12:14 implies a previous carrying away. Yet more Zechariah took his imagery of the future restoration of Jerusalem, from its condition in his own time. "It shall be lifted up and inhabited in its place from Benjamin's gate unto the place of the first gate, unto the corner-gate, and from the tower of Hananeel unto the king's winepresses" Zechariah 14:10. "The gate of Benjamin" is doubtless "the gate of Ephraim," since the road to Ephraim lay through Benjamin but the gate of Ephraim existed in Nehemiah's time Nehemiah 8:16 Nehemiah 12:39, yet was not then repaired, as neither was the tower of Hananeel Nehemiah 3:1, having been left, doubtless, at the destruction of Jerusalem, being useless for defense, when the wall was broken down. So at the second invasion the Romans left the three impregnable towers, of Hippicus, Phasaelus, and Mariamne, as monuments of the greatness of the city which they had destroyed. Benjamin's gate, the corner gate, the tower of Hananeel, were still standing "the king's winepresses" were naturally uninjured, since there was no use in injuring them but "the first gate" was destroyed, since not itself but "the place" of it is mentioned.

The prophecy of the victory over the Greeks fits in with times when Assyria or Chaldaea were no longer the instruments of God in the chastisement of His people. The notion that the prophet incited the few Hebrew slaves, sold into Greece, to rebel against their masters, is so absurd, that one wonders that any one could have ventured to forge it and put it upon a Hebrew prophet .

Since, moreover, all now, who sever the six last chapters from the preceding, also divide these six into two halves, the evidence that the six chapters are from one author is a separate ground against their theory. Yet, not only are they connected by the imagery of the people as the flock of God Zechariah 9:16 Zechariah 10:3, whom God committed to the hand of the Good Shepherd Zechariah 11:4-14, and on their rejecting Him, gave them over to an evil shepherd Zechariah 11:15-17 but the Good Shepherd is One with God Zechariah 11:7-12 Zechariah 13:7. The poor of the flock, who would hold to the Shepherd, are designated by a corresponding word.

A writer has been at pains to show that two different conditions of things are foretold in the two prophecies. Granted. The first, we believe, has its foreground in the deliverance during the conquests of Alexander, and under the Maccabees, and leads on to the rejection of the true Shepherd and God's visitation on the false. The later relates to a later repentance and later visitation of God, in part yet future. By what law is a prophet bound down to speak of one future only?

For those who criticize the prophets, resolve all prophecy into mere "anticipation" of what might, or might not be, denying to them all certain knowledge of any future, it is but speaking plainly, when they imagine the author of the three last chapters to have "anticipated" that God would interpose miraculously to deliver Jerusalem, then, when it was destroyed. It would have been in direct contradiction to Jeremiah, who for 39 years in one unbroken dirge predicted the evil which should come upon Jerusalem. The prophecy, had it preceded the destruction of Jerusalem, could not have been earlier than the reign of the wretched Jehoiakim, since the mourning for the death of Josiah is spoken of as a proverbial sorrow of the past. This invented prophet then would have been one of the false prophets, who contradicted Jeremiah, prophesying good, while Jeremiah prophesied evil who encouraged Zedekiah in his perjury, the punishment whereof Ezekiel solemnly denounced Ezekiel 13:10-19, prophesying his captivity in Babylon as its penalty he would have been one of those, of whom Jeremiah said that they spake lies Jeremiah 14:14 Jeremiah 23:22 Jeremiah 27:15 Jeremiah 28:15 Jeremiah 29:8-9 in the name of the Lord. It was not "anticipation" on either side.

It was the statement of those who spoke more certainly than we could say, "the sun will rise tomorrow." They were the direct contradictories of one another. The false prophets said, "the Lord hath said, Ye shall have peace" Jeremiah 8:11 Jeremiah 23:17 the true, "they have said, 'Peace, peace,' when there is no peace" Ezekiel 13:2-10 the false said, "sword and famine shall not be in the land" Jeremiah 14:15 the true "By sword and famine shall their prophets be consumed" the false said, "ye shall not serve the king of Babylon thus saith the Lord, even so will I break the yoke of Nebuchadnezzar, king of Babylon, from the neck of all nations within the space of two full years" Jeremiah 27:9-14 Jeremiah 28:11 the true, "Thus saith the Lord of hosts, Now have I given all these lands into the hand of Nebuchadnezzar the king of Babylon, My servant, and all nations shall serve him, and his son and his son's son" Jeremiah 27:4, Jeremiah 27:6-7. The false said, "I will bring again to this place Jeconiah, with all the captives of Judah, that went into Babylon, for I will break the yoke of the king of Babylon" Jeremiah 28:4 the true, "I will cast thee out and the mother that bare thee, into another country, where ye were not born, and there ye shall die. But to the land, whereunto they desire to return, thither they shall not return" Jeremiah 22:26-27. The false said "The vessels of the Lord's house shall now shortly be brought again from Babylon" Jeremiah 27:16 the true "the residue of the vessels that remain in this city, - they shall be carried to Babylon" Jeremiah 27:19-22.

If the writer of the three last chapters had lived just before the destruction of Jerusalem in those last reigns, he would have been a political fanatic, one of those who, by encouraging rebellion against Nebuchadnezzar, brought on the destruction of the city, and, in the name of God, told lies against God. "That which is most peculiar in this prophet," says one , "is the uncommon high and pious hope of the deliverance of Jerusalem and Judah, notwithstanding all visible greatest dangers and threatenings. At a time when Jeremiah, in the walls of the capital, already despairs of any possibility of a successful resistance to the Chaldees and exhorts to tranquility, this prophet still looks all these dangers straight in the face with swelling spirit and divine confidence, holds, with unbowed spirit, firm to the like promises of older prophets, as Isaiah 29, and anticipates that, from that very moment when the blind fury of the destroyers would discharge itself on the sanctuary, a wondrous might would crush them in pieces, and that this must be the beginning of the Messianic weal within and without."

Zechariah 14 is to this writer a modification of those anticipations. In other words there was a greater human probability, that Jeremiah's prophecies, not his, would be fulfilled: yet, he cannot give up his sanguineness, though his hopes had now become fanatic. This writer says on Zechariah 14 , "This piece cannot have been written until somewhat later, when facts made it more and more improbable, that Jerusalem would not any how be conquered, and treated as a conquered city by coarse foes. Yet, then too, this prophet could not yet part with the anticipations of older prophets and those which he had himself at an earlier time expressed: so boldly, amid the most visible danger, he holds firm to the old anticipation, after that the great deliverance of Jerusalem in Sennacherib's time Isaiah 37 appeared to justify the most fanatic hopes for the future, (compare Psalm 59). And so now the prospect moulds itself to him thus, as if Jerusalem must indeed actually endure the horrors of the conquest, but that then, when the work of the conquerors was half-completed, the great deliverance, already suggested in that former piece, would come, and so the Sanctuary would, notwithstanding, be wonderfully preserved, the better Messianic time would notwithstanding still so come."

It must be a marvelous fascination, which the old prophets exercise over the human mind, that one who can so write should trouble himself about them. It is such an intense paradox, that the writing of one convicted by the event of uttering falsehood in the name of God, incorrigible even by the thickening tokens of God's displeasure, should have been inserted among the Hebrew prophets, in times not far removed from those whose events convicted him, that one wonders that anyone should have invented it, still more that any should have believed in it. Great indeed is "the credulity of the incredulous."

And yet, this paradox is essential to the theories of the modern school which would place these chapters before the captivity. English writers, who thought themselves compelled to ascribe these chapters to Jeremiah, had an escape, because they did not bind down prophecy to immediate events. Newcome's criticism was the conjectural criticism of his day i. e. bad, cutting knots instead of loosing them. But his faith, that God's word is true, was entire. Since the prophecy, placed at the time where he placed it, had no immediate fulfillment, he supposed it, in common with those who believe it to have been written by Zechariah, to relate to a later period. That German school, with whom it is an axiom, "that all definite prophecy relates to an immediate future," had no choice but to place it just before the destruction of the temple by the Chaldees, or its profanation by Antiochus Epiphanes and those who placed it before the Captivity, had no choice, except to believe, that it related to events, by which it was falsified.

Nearly half a century has passed, since a leading writer of this school said , "One must own, that the division of opinions as to the real author of this section and his time, as also the attempts to appropriate single oracles of this portion to different periods, leave the result of criticism simply "negative" whereas on the other hand, the view itself, since it is not yet carried through exegetically, lacks the completion of its proof. It is not till criticism becomes "positive," and evidences its truth in the explanation of details, that it attains its completion which is not, in truth, always possible." Hitzig did what he could, "to help to promote the attainment of this end according to his ability." But although the more popular theory has of late been that these chapters are to be placed before the captivity, the one portion somewhere in the reigns of Uzziah, Jotham, Ahaz, or Hezekiah the other, as marked in the chapters themselves, after the death of Josiah there have not been wanting critics of equal repute, who place them in the time of Antiochus Epiphanes. Yet, criticism which reels to and fro in a period of near 500 years, from the earliest of the prophets to a period a century after Malachi, and this on historical and philological grounds, certainly has come to no definite basis, either as to history or philology.

Rather, it has enslaved both to preconceived opinions and at last, as late a result as any has been, after this weary round, to go back to where it started from, and to suppose these chapters to have been written by the prophet whose name they bear .

It is obvious that there must be some mistake either in the tests applied, or in their application, which admits of a variation of at least 450 years from somewhere in the reign of Uzziah (say 770 b.c.) to "later than 330 b.c."

open mine eyes upon … Judah—to watch over Judah's safety. Heretofore Jehovah seemed to have shut His eyes, as having no regard for her.

blindness—so as to rush headlong on to their own ruin (compare Zec 14:12, 13).

I will smite every horse : horses are of very great use in wars they were the main strength of Antiochus Epiphanes, his best preparations. With astonishment a dull, sottish fear and perplexity.

And his rider with madness an impotency of mind both in the understanding, which is folly and imprudence, and in the will and resolution, which is either cowardice or unconstancy, like madmen that neither know how to resolve or act. God will turn all their counsel into foolishness, their strength into weakness, their courage into fear, and so overturn them all.

I will open mine eyes upon the house of Judah a while I seemed as one that slept or winked at the proceedings of my church’s enemies, yet now I will open mine eyes, and see all that is going forward against them, and I will watch over my people for good against their enemies, to confound and destroy them and their enterprises: this eye of God open upon his people is his wise, powerful, gracious providence for them, Psalm 31:22 Jeremiah 24:6 .

I will smite every horse of the people with blindness all their warriors in their projecting and consults shall be as full of improvidence, and have as little foresight, as a stark blind man hath of sight to see by.

and I will open mine eyes upon the house of Judah which phrase is sometimes used, as expressive of the wrath of God against his enemies, Amos 9:4 and, if the house of Judah signifies the same as Judah, joined with the nations of the earth in the siege, Zechariah 12:2, it must be so understood here but rather it seems to be different, and to intend those who will inhabit other parts of Judea, and who will be truly the people of God, Jews not only literally, but spiritually and so is to be interpreted in a good sense, of the divine love to them, care of them, and protection over them see Job 14:13 and so the Targum paraphrases it,

"and upon those of the house of Judah, I will reveal my power to do them good:''

and will smite every horse of the people with blindness: that is, every rider of them, either with blindness of mind or body, or both. It may be, as the former smiting, mentioned in the beginning of the verse, respects the mind, this may regard the body so that they shall not see their way, and their hands shall not perform their enterprise.

4 . astonishment ] This and the two following words, madness, blindness , occur together also in Deuteronomy 28:28, in a description of God’s judgments upon Israel, as here upon the armies that gather against Jerusalem.

## Temple Fork Outfitters

**The Axiom II-X was designed for the intermediate to advanced fly angler seeking to maximize accuracy at distance. Based on the fast action of our renowned TiCrX, we used our highest modulus material and Axiom technology to redefine performance in an extremely powerful fly rod. Unlike other “stiff” rods, the Axiom II-X delivers both the energy necessary for long casts and the incredible tracking and recovery which results in accuracy at distance. If it comes down to one cast, one perfect long cast, this is the fishing tool to do the job.**TFO’s patented and exclusive Axiom technology embeds a double-helix of Kevlar within the blank. The superior tensile strength of the Kevlar acts to buttress the rod’s carbon fiber matrix in compression. The result is that Axiom series fly rods stabilize faster and smoother, absorb shock better and comfortably tolerate over-loading. The angler benefits because Axiom technology virtually eliminates the ability to over power the rod when casting. Bottom line – whether you carry more line in the air or push the rod to the limit, you won’t feel any mushiness – What you will feel is line ripping out of your hand as it launches.

The Axiom II-X series is constructed with high modulus carbon fiber material and an embedded double-helix of Kevlar within the blank all finished in a satin sky blue. The series features premium quality cork handles with burl accents, anodized aluminum up-locking reel seats with carbon fiber inserts. All eight models feature alignment dots color coded by line weight,

*RECOIL*guides by REC and ultra-lightweight chromium-impregnated stainless-steel snake guides. All Axiom II-X rods are packaged in a labeled rod sock and rod tube.The Axiom II-X series delivers exceptional casting performance and efficiency that when combined with TFO’s no-fault lifetime warranty make them the perfect choice when distance and accuracy are the price of a life time trophy.

## 12.4: Axiom I

**Axiom is a dynamic infrastructure framework**to efficiently work with multi-cloud environments, build and deploy repeatable infrastructure focussed on offensive and defensive security.Axiom works by pre-installing your tools of choice onto a 'base image', and then using that image to deploy fresh instances. From there, you can connect and instantly gain access to many tools useful for both bug hunters and pentesters. With the power of immutable infrastructure, most of which is done for you, you can just spin up 15 boxes, perform a distributed nmap/ffuf/screenshotting scan, and then shut them down.

Because you can create many disposable instances very easily, axiom allows you to distribute scans of many different tools including amass aquatone arjun assetfinder dalfox dnsgen dnsx feroxbuster fff ffuf findomain gau gobuster gospider gowitness hakrawler httprobe httpx jaeles kiterunnter masscan massdns meg naabu nmap nuclei paramspider puredns rustscan s3scanner shuffledns & subfinder. Once installed and setup, you can distribute a scan of a large set of targets across 10-15 instances within minutes and get results extremely quickly. This is called axiom-scan.

Axiom supports several cloud providers, eventually, axiom should be completely cloud agnostic allowing unified control of a wide variety of different cloud environments with ease. Currently, DigitalOcean, IBM Cloud, Linode and Azure are officially supported providers. Google Compute is partially implemented and AWS is on the roadmap. If you would like prioritization of a feature or provider implementation, please contact me @pry0cc on Twitter and we can discuss :)

The original and best supported provider for Axiom is Digital Ocean! If you're signing up for a new Digital Ocean account, please use my link!

Our third provider for axiom! Please use this link for $20 free credit on Linode :)

Installation - Easy Install

You will also need to install the newest versions of all packages sudo apt dist-upgrade and curl, which is not installed by default on Ubuntu 20.04, if you get a "command not found" error, run sudo apt update && sudo apt install curl .

Run the following curl command, as your standard user, not as root.

If you have any problems with this installer, please refer to Installation.

In this demo (sped up out of respect for your time ) ), we show how easy it is to initialize and ssh into a new instance.

If you like Axiom and it saves you time, money or just brings you happy feelings, please show your support through sponsorship! Click the little sponsor button in the header and sponsor for as little as $1 per month :)

Or buy me a coffee to keep me powered :)

Sponsored By SecurityTrails!

We are lucky enough to be sponsored by the awesome SecurityTrails! Sign up for your free account here!

Operating Systems Supported

OS Supported Easy Install Tested Ubuntu Yes Yes Ubuntu 20.04 Kali Yes Yes Kali 2020.4 Debian Yes Yes Debian 10 Windows Yes Yes WSL w/ Ubuntu MacOS Yes No MacOS 10.15 Arch Linux Yes No Yes We've had some really fantastic additions to axiom, great feedback through issues, and perseverence through our heavy beta phase!

A list of all contributors can be found here, thank you all!

The logo was made by our amazing s0md3v! Thank you for making axiom look sleek as hell! Really beats my homegrown logo :)

- amass
- anew
- anti-burl
- aquatone
- assetfinder
- dalfox
- dirb
- dnsprobe
- dnsvalidator
- docker
- fbrobe
- feroxbuster
- ffuf
- gau
- getjs
- gf
- gobuster
- Golang (setup, path configured, latest version)
- gowitness
- hakrawler
- httprobe
- jq
- kxss
- masscan
- massdns
- metasploit
- mosh
- nmap
- oh-my-zsh
- openvpn
- Paramspider
- projectdiscovery chaos
- projectdiscovery chaos-client
- projectdiscovery httpx
- projectdiscovery naabu
- projectdiscovery nuclei
- projectdiscovery shuffledns
- proxychains w/ Tor setup
- SecLists
- sn0int
- SQLMap
- subfinder
- subgen
- subjack
- tmux
- urlprobe
- waybackurls
- zdns
- zmap

And many more! Do you want to add a package to axiom? Let me know!

## AXIOM IS YOUR PARTNER IN

The most skilled tailors make things fit you in a way that no one else can. They care more about your confidence and success than which brands they’re selling you. For over 35 years that’s how we’ve served our customers. Whatever the challenge, we’ll get it right.

#### Space Utilization

Take advantage of your existing space by finding new ways to increase storage capacity without the need for a costly building expansion.

#### Process Efficiency

Become more lean and efficient in your operations, improving productivity, order fulfillment accuracy and speed, inventory organization, and more.

#### Safety & Ergonomics

Create better ways to improve the safety of your facility and your workers by investing in ergonomic-friendly solutions and safety products.

A landmark agreement between Axiom Space and SpaceX confirms Axiom's next three planned missions to the International Space Station will fly on SpaceX's Dragon, in addition to Ax-1.

The growing partnership between Axiom and SpaceX - the industry leaders in human spaceflight and in orbital services and launch, respectively - solidifies the nascent commercial human spaceflight market

The missions, both managed and launched by private companies, are a validation of NASA's Commercial Crew strategy to enable a commercial marketplace in low-Earth orbit

**HOUSTON — 2 June 2021**Signed, sealed, and delivered: the commercialization of low-Earth orbit is in full swing.

Axiom Space revealed Wednesday that it has finalized a deal with SpaceX for three additional Dragon flights, on which Axiom would fly its proposed private crews on its next three fully commercial missions to the International Space Station. The landmark agreement between the industry leaders in human spaceflight as well as launch and orbital services, respectively, ensures the nascent commercial human spaceflight market’s growth will subsist.

“We are beyond excited to build upon our partnership with Axiom to help make human spaceflight more accessible for more people,” said SpaceX President & COO Gwynne Shotwell. “A new era in human spaceflight is here.”

Developed by SpaceX as part of NASA’s Commercial Crew program, the Dragon spacecraft has already flown three successful human spaceflight missions to the ISS. Those flights – Demo-2, Crew-1, and Crew-2 – were NASA missions carrying government astronauts from the ISS partner agencies.

In a validation of NASA’s strategy to support commercial development of human spaceflight capability in hopes of fostering a marketplace, Axiom’s planned missions would mark the first private crews to make the same trip.

"Axiom was founded on a vision of lasting commercial development of space,” Axiom President & CEO Michael Suffredini said. “We are on track to enable that future by managing the first-ever private missions to the ISS as a precursor to our development of the world’s first commercial space station. SpaceX has blazed the trail with reliable, commercial human launch capability and we are thrilled to partner with them on a truly historic moment."

Ax-1, Axiom’s historic first private ISS mission, has already been approved by NASA and targeted for launch to the ISS no earlier than Jan. 2022, also aboard Dragon as a result of a deal the companies signed in March 2020. Axiom last week revealed legendary astronaut Peggy Whitson and champion GT racer John Shoffner would serve as commander and pilot on its proposed Ax-2 mission – now confirmed to be a Dragon flight.

Axiom previously entered into a broad agreement with NASA enabling it to fly private astronaut missions to the space station and will compete to fly each as the agency opens opportunities. All-inclusive Axiom missions include training, provisions, and operational management, are commanded by experienced astronauts, and are built around the crew’s preferred scientific research and educational programs.

Axiom’s “precursor missions” prepare the way for the launch and integration to the ISS Harmony node of the Axiom Station modules beginning in 2024. By 2028, Axiom Station will be ready to detach and operate as the ISS’ privately owned successor, forming the core layer of infrastructure in orbit for years to come.

The growing partnership between Axiom and SpaceX thus lays the groundwork for a long-term destination for Dragon, more humans in space, and a burgeoning economy in low-Earth orbit – realizing a dream long-held by advocates of commercial space.

## Work with the global leader in high-caliber and diverse legal talent.

Explore our network to find the right legal talent, with the right level of experience and the most relevant expertise.

##### Insights

#### The Real Costs of Legal Hiring

Why engaging flexible talent is the cost-efficient approach to building high-performing legal teams.

##### Case Study

#### Partnership with Yext

How Axiom helped Yext deliver a regional expertise solution to support their international growth.

“By partnering with Axiom, we get the quality, flexibility and expertise we would normally find at a top law firm but delivered in a way that’s much more integrated with our business. It’s enabled us to grow faster and reduce risk.”

"Axiom lawyers have the same acumen and capability as large law firm lawyers or members of my in-house team, and they sit with us to understand how the business operates"

“I never would have thought external providers could work so efficiently and seamlessly with us and become part of the legal team.”

“We were able to accomplish more as a department, save over $400,000 on our legal spend, and maintain flexibility as our needs changed.”

## Watch the video: Axiom концепция (December 2021).