# Discrete Math Set Theory Question

Chandasouk
Let A, B, and C be sets. Show that

a) (A-B) - C $\subseteq$ A - C
b) (B-A) $\cup$ (C-A) = (B $\cup$ C) - A

I am using variable x to represent an element.

Part A)

I rewrote (A-B) - C as (x$\in$A ^ x$\notin$B) - C

I think this could be rewritten as
(x$\in$A ^ x$\notin$B) ^ x$\notin$ C

A-C can be rewritten as (x $\in$ A ^ x $\notin$ C)

The original statement can be rewritten as

x$\in$A$\cap$~B$\cap$~C $\subseteq$ x$\in$A$\cap$~C

where ~ represents negation.

However, for the LHS to be a subset of the RHS, all elements of the LHS should be an element of RHS but since the LHS has ~B, I don't think that it is a subset?

I have no idea how to show part B so any help would be great.

## Answers and Replies

Mentor
when you draw a venn diagram you'll see that set A-C contains the intersection A,B and C but that A-B-C does not. perhaps you use that in your proof. I don't think venn diagrams can be used in proofs but they do help visualize the situation.