How Do You Prove Set Theory Relations and Operations?

[mbcsantin]

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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

Homework Statement

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

Homework Equations

none.

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.

 

[vacuum]

First of all, great job on attempting these proofs! I will go through each one and provide some feedback and corrections where needed.

1) Your first proof looks good, but there are a few small errors. In the first case, you say that "x is an element of A and x is an element of B." This should actually be "x is an element of A and x is an element of A n B." Also, in Case 2, you say that "x is an element of B" but this should be "x is an element of A n B." Finally, in the last line, you say "A contained A n B" but this should be "A contains A n B."

2) In this proof, you seem to be assuming that A is a proper subset of U, but this is not stated in the problem. You also do not use the fact that A = Universe and B = Universe in your proof. Instead, you should start by assuming that A n B = Universe and then show that A = Universe and B = Universe.

3) This counterexample does not work because you are assuming that A is the empty set, but in the problem, it says that A, B, and C are all sets in the universe U. So A cannot be the empty set. Instead, you can use the sets {1, 2, 3} for A, {2, 3, 4} for B, and {3, 4, 5} for C to show that the statement is false. A union C contains B union C, but A does not contain B.