1. ## proof

if A union B = A union C, then B=C.

proof: Assume that A union B = A union C. Suppose x is in B. Then x is in A U B. By the assumption x is also in A U C. But we did not assume x was in A so x must be in C. Thus,we have shown B is a subset of C and by a parallel argument C is a subset of B

is this right

if A union B = A union C, then B=C.

proof: Assume that A union B = A union C. Suppose x is in B. Then x is in A U B. By the assumption x is also in A U C. But we did not assume x was in A so x must be in C. Thus,we have shown B is a subset of C and by a parallel argument C is a subset of B

is this right
Not quite. Let $A=\{1, 2, 3\}$, $B=\{3,4\}$. Can you perhaps see the problem in your proof by looking at these examples?

However, you can quite easily tweak your proof to make it valid, taking into account the problem I've hinted at.

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