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Let X and Y be non-empty sets. If A_1 and A_2 are subsets of X, and B_1 and B_2 subsets of Y. Show that (A_1×B_1) intersection (A_2×B_2) = (A_1 intersection A_2)×(B_1 intersection B_2).

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Topology
Sets and Functions (XLI)
Functions

Let X and Y be non-empty sets. If A_1 and A_2 are subsets of X, and B_1 and B_2 subsets of Y.

Show that

(A_1×B_1) intersection (A_2×B_2) = (A_1 intersection A_2)×(B_1 intersection B_2).

See the attached file.

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Solution Preview

Topology
Sets and Functions (XLI)
...

Solution Summary

This solution is comprised of a detailed explanation of the properties of the products of sets.
It contains step-by-step explanation of the following problem:

Let X and Y be non-empty sets. If A_1 and A_2 are subsets of X, and B_1 and B_2 subsets of Y.
Show that
(A_1×B_1) intersection (A_2×B_2) = (A_1 intersection A_2)×(B_1 intersection B_2).

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