Is My Set Theory Proof Correct?

[mbcsantin]

I tried to do the questions but I am just not sure if i did it right. id appreciate if you can check my work and let me know what changes i have to make. thanks

the symbol "n" means "intersect"
U for Union

Question: Prove A contained B iff A n B = A

Answer:
(=>) Assume A contained B

Let x is an element of A, since A n A = A, x is an element of A and x is an element of B

Case 1: x is an element of A: Since A contained B, x is an element of B so
x is an element of A n B

Case 2: x is an element of B: If x is an element of B then
x is an element of (A n B)

Hence x is an element of A n B

This shows A contained A n B

(<=) Assume A n B = A then

A’=A’UA
= A’ U (A n B)
=(A’UA) n (A’U B)
= empty set n A’ U B
= A’ U B

Hence
Universe = A’ U B




Question: Prove the following: For any sets A, B, C in a universe U:

a) A n B = Universe iff A = Universe and B = Universe

Answer: Suppose A n B = U and suppose that A is a proper subset of U then
x is an element of B but
x is not an element of A n B since x is not an element of A




Question: Prove or find counterexamples. For any sets A, B, C in a universe U:

a) if A union C contained B union C then A contained B

Answer: Let A be the empty set, and let B = C
Then A union C = B and
B union C = B so,
A union C contains B union C, but A does not contain B because A is the empty set and B is not.