For any sets A, B, C, D, prove that (A×B) ∩ (C×D) = (A∩C) × ( B∩D) ​

Question

For any sets A, B, C, D, prove that

(A×B) ∩ (C×D) = (A∩C) × ( B∩D) ​

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Ariana 2 months 2021-10-04T21:57:16+00:00 1 Answer 0 views 0

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    2021-10-04T21:58:44+00:00

    Answer:

    Show that (A×B)∩(C×D)⊆(A∩C)×(B∩D) first, and then show the other inclusion ⊇.

    For ⊆: Let z=(x,y)∈(A×B)∩(C×D). That means (x,y)∈A×B and (x,y)∈C×D. Now finish it and conclude that z=(x,y)∈(A∩C)×(B∩D).

    For ⊇: Let z=(x,y)∈(A∩C)×(B∩D). That means x∈A∩C and y∈B∩D. Now conclude.

    Step-by-step explanation:

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