Kenneth Rosen Edition 6th Exercise 2.2 Question 24 (Page No. 131)

Question

Q-24 Let A,B and C be sets. Show that (A - B) -C = (A - C) - (B - C)

and I think ( A - B) -C = (A - C) - B

Comments

is it correct?

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yeah  aditi19

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Yes, that's correct. You can also verify by giving the number. It's best way to verify.

Answer 1

If $x \in (A-B)$ then it means that $(x \in A) \land (x \notin B)$ , We follow this basics to prove the above theorem

R.H.S. = $(A - C) - (B - C)$

           = $ (x \in (A - C)) \land (x \notin (B - C)) $

           = $ (x \in A) \land (x \notin C) \land (x \notin B) \land (x \notin C) $

           = $ (x \in A) \land (x \notin B) \land (x \notin C) \land (x \notin C) $

           = $ (x \in A) \land (x \notin B) \land (x \notin C) $

           = $ (x \in (A - B) ) \land (x \notin C) $

           = $ (x \in ((A - B) - C)) $

           = $ (A - B) - C $

Comments

@manojk ?

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Here , You should have to do the same for LHS to RHS also.. Because 2 sets A and B are equal iff A is a subset of B and B is a subset of A.

To prove it, we can use Venn Diagram also.