Let A, B, C be arbitrary finite sets from the same universal set U. - (a) Is it true that A - B C (A - B) – (B − C)? If "yes", then prove "rigorously"; if "no", then show a concrete counterexample by specifying sets A, B, C where this subset relation does not hold. (To prove an expression of the form MCN "rigorously", you need to consider an arbitrary element x from M and show that x E N.) (b) Is it true that (A - B = A - C) ⇒ (B = C)? If "yes", then prove "rigorously"; if "no", then show a concrete counterexample by specifying sets A, B, C where this implication does not hold.

Let A, B, C be arbitrary finite sets from the same universal set U. - (a) Is it true that A - B C (A - B) – (B − C)? If "yes", then prove "rigorously"; if "no", then show a concrete counterexample by specifying sets A, B, C where this subset relation does not hold. (To prove an expression of the form MCN "rigorously", you need to consider an arbitrary element x from M and show that x E N.) (b) Is it true that (A - B = A - C) ⇒ (B = C)? If "yes", then prove "rigorously"; if "no", then show a concrete counterexample by specifying sets A, B, C where this implication does not hold.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.7: Relations

Problem 20E: Give an example of a relation R on a nonempty set A that is symmetric and transitive, but not...

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