Find the mistakcs in the following proof Theorem For all se

Find the mistakc(s) in the following \" proof\' Theorem: For all sets A ami D Prove, using elements, as above, For all sets A, B. and

Solution

Suppose A is a subset of B. That is to say, if x is in A then x is in B.

Claim: A U C is a subset of B U C.

Proof: Suppose x is in A U C. Then x is in A or x is in C. In the case that x is in A, since A is a subset of B, it follows that x is in B; and hence x is in B U C. On the other hand, if x is in C, then it follows immediately that x is in B U C. In either case, if x is in A U C, then x is in B U C. Therefore, A U C is a subset of B U C.