Mathematics: proof with sets

Question

A) Prove that given sets S and T, we have S⊆T if and only if S=S∩T

B) If S∩T=T∪S, then S=T

C) If SxT=TxS and both T and S are nonempty, then S=T

Answer 1

A)

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Given S is a subset of T then S = S intersect T

Let x be an element of S.

Then x is an element of T since S is a subset of T.

As a result x is an element of BOTH S and T.

Therefore x is an element of S intersect T.

[end of sub-proof]

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Given S = S intersect T Then S is a subset of T.

Let x be an element of S intersect T

Then x is an element of S and x is an element of T.

So then, is x is an element of S, then x is an element of T.

Therefore, S is a subset of T.

B)

For any element x in S intersect T, x is also an

element of S union T.

Therefore x is an element of S and T, so S=T

C)

Let s be an element of S and t be an element of T.

s*t = S x T = T x S = t*s are the same element.

THerefore S=T