3. An "interpretation" for a logical statement consists of a domain D (any non-empty set of elements) and a meaning for each predicate symbol. For example, D= {1,2} and P(x): "> 0" is an interpre- tation for the statement Vr ED, P(x) (in this case, one that happens to make the statement true). For each statement below, provide one interpretation under which the statement is true and another interpretation under which the statement is false if either case is not possible, explain why clearly and concisely. You may reuse examples if you wish. (a) (b) (c) xD, Vy € D, P(x, y) Vr ED, Vy ED, P(x,y) P(y, x) P(y, z)] AN ED, Vy ED,-P(x, y)] D.Q(x)] Nx € D, P(x)]

3. An "interpretation" for a logical statement consists of a domain D (any non-empty set of elements) and a meaning for each predicate symbol. For example, D= {1,2} and P(x): "> 0" is an interpre- tation for the statement Vr ED, P(x) (in this case, one that happens to make the statement true). For each statement below, provide one interpretation under which the statement is true and another interpretation under which the statement is false if either case is not possible, explain why clearly and concisely. You may reuse examples if you wish. (a) (b) (c) xD, Vy € D, P(x, y) Vr ED, Vy ED, P(x,y) P(y, x) P(y, z)] AN ED, Vy ED,-P(x, y)] D.Q(x)] Nx € D, P(x)]

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter2: The Integers

Section2.1: Postulates For The Integers (optional)

Problem 18E: In Exercises , prove the statements concerning the relation on the set of all integers. 18. If ...

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