# Elementary proof check/help

## Homework Statement

For all sets A, B, and C, prove or provide a counterexample the following statements.
(A $\setminus$ B) $\cap$ (C $\setminus$ B) = A $\setminus$ (B $\cup$ C).

N/A

## The Attempt at a Solution

I went ahead and said it was false, and provided a counter example. I'm new to this and just wanna make sure my thought process was correct and the statement is indeed false.
Counterexample:
Let A = {1,2,3}, B = {2,3}, and C = {1,2}.
A $\setminus$ B = {1} and C $\setminus$ B = {1}. Then (A $\setminus$ B) $\cap$ (C $\setminus$ B) = {1}.
B $\cup$ C = {1,2,3}. Then A $\setminus$ (B $\cup$ C) = {0}. Since {1} ≠ {0}, the statement is false.

It's more the last part, i.e. "Then A $\setminus$ (B $\cup$ C) = {0}" I want to make sure is correct. Thanks for your help!

jbunniii
Homework Helper
Gold Member

## Homework Statement

For all sets A, B, and C, prove or provide a counterexample the following statements.
(A $\setminus$ B) $\cap$ (C $\setminus$ B) = A $\setminus$ (B $\cup$ C).

N/A

## The Attempt at a Solution

I went ahead and said it was false, and provided a counter example. I'm new to this and just wanna make sure my thought process was correct and the statement is indeed false.
Counterexample:
Let A = {1,2,3}, B = {2,3}, and C = {1,2}.
A $\setminus$ B = {1} and C $\setminus$ B = {1}. Then (A $\setminus$ B) $\cap$ (C $\setminus$ B) = {1}.
B $\cup$ C = {1,2,3}. Then A $\setminus$ (B $\cup$ C) = {0}. Since {1} ≠ {0}, the statement is false.
This is almost right. Your notation {0} is incorrect. That refers to a set containing one element, the number 0. What you want is $\emptyset$, the empty set.

Aside from that your counterexample looks fine.

I meant the empty set, but I guess 0 would be element and not empty. Thanks for telling the difference, I won't make that same mistake again.

HallsofIvy