Set theory - counterexamples, true/false!

[MogiYagi]

Hi!

First off, thank you for the help I recieved yesterday, it greatly helped me.
Now I have another problem.

I'm supposed to prove if these statements are correct:

a) Bc = (A ∪ C) \ B

c) Ac ∪ Bc ∪ Cc = (A ∪ B ∪ C)c

What I did was I broke everything down into right- and left tables, and I saw disrepancy which proved that both the answers are not correct. This was not enough though. What I have to do is:

Counterexamples must contain elements , and a basic amount U, and it should be clear what the contradiction arises from and what you draw for conclusion.

How do I give an counterexample to prove that this is not correct? Please help!

Greetings!

 

[HallsofIvy]

If a statement about sets is NOT TRUE for all sets, then there must exist some specific sets for which it is not true. A counter example is such a statement. Since you only have to give one, try very simple sets!

For the first one, for example, let the universal set, U, (which is what I think you mean by "basic amount") be the alphabet: {a, b, c, d, ..., x, y, z}, A= {a, b, c}, B= {a, b}, and C= {d} (made up pretty much at random).

What is \(\displaystyle (A\cup C)\ B\)? What is \(\displaystyle B^c\)? If they are not the same then this is a counter-example to "\(\displaystyle (A\cup C)\B = B^c\)".