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I tried to do the questions but im 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

1. The problem statement, all variables and given/known data

1) Prove A contained B iff A n B = A

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

A n B = Universe iff A = Universe and B = Universe

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

if A union C contained B union C then A contained B

2. Relevant equations

none.

3. The attempt at a solution

1) (=>) 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

2) 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

3) 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.

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

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# Homework Help: Help, set theory proving

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