How to prove this set problem?

If A union( B intersection C), then (A union B) intersection (A union C)


I will use + for union, & for intersection

A+(B&C) = (A+B)&(A+C)


let x be in A+(B&C)

then x is in A or x is in both B and C

A is a subset of A+B and also of A+C, so that if x is in A, it is in both A+B and A+C, so it is in the intersection. if x is in B and in C, then x is in both A+B and A+C, so it is in the intersection.

therefore A+(B&C) is a subset of (A+B) & (A+C)

now let y be in (A+B)&(A+C)

if y in A, then y in A + (B&C). so suppose y is not in A

then, since y in A+B, y is in B, and since y in A+C, y is in C, therefore y (if not in A) is in B&C

therefore y is in A + (B&C)

therefore (A+B)&(A+C) is a subset of A+(B&C)

therefore these sets are equal


rither use venn diagram or make use of taking an arbitrary element method.


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