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Discrete Math Set Theory Question

  • Thread starter Chandasouk
  • Start date
  • #1
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Let A, B, and C be sets. Show that

a) (A-B) - C [itex]\subseteq[/itex] A - C
b) (B-A) [itex]\cup[/itex] (C-A) = (B [itex]\cup[/itex] C) - A

I am using variable x to represent an element.

Part A)

I rewrote (A-B) - C as (x[itex]\in[/itex]A ^ x[itex]\notin[/itex]B) - C

I think this could be rewritten as
(x[itex]\in[/itex]A ^ x[itex]\notin[/itex]B) ^ x[itex]\notin[/itex] C

A-C can be rewritten as (x [itex]\in[/itex] A ^ x [itex]\notin[/itex] C)

The original statement can be rewritten as

x[itex]\in[/itex]A[itex]\cap[/itex]~B[itex]\cap[/itex]~C [itex]\subseteq[/itex] x[itex]\in[/itex]A[itex]\cap[/itex]~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.

Homework Statement





Homework Equations





The Attempt at a Solution

 

Answers and Replies

  • #2
11,658
5,231
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.
 

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