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For each of the following relations, determine whether the relation is: Reflexive. Transitive. A partial order. A strict order. Anti-reflexive. Symmetric. Anti-symmetric. An equivalence relation. Justify all your answers. a. Ris a relation on the set of all people such that (a, b) e R if and only if a and b have a common grandparent. b. Ris a relation on Z such that (x, y) E R if and only if |x – y| < 2. c. Ris a relation on Z* such that (x, y) € R if and only if x is divisible by y. Hint: An integer x is divisible by an integer y with y # 0 if and only if there exists an integer k such that x = yk. 4.

For each of the following relations, determine whether the relation is: Reflexive. Transitive. A partial order. A strict order. Anti-reflexive. Symmetric. Anti-symmetric. An equivalence relation. Justify all your answers. a. Ris a relation on the set of all people such that (a, b) e R if and only if a and b have a common grandparent. b. Ris a relation on Z such that (x, y) E R if and only if |x – y| < 2. c. Ris a relation on Z* such that (x, y) € R if and only if x is divisible by y. Hint: An integer x is divisible by an integer y with y # 0 if and only if there exists an integer k such that x = yk. 4.

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter1: Fundamentals
Section1.7: Relations
Problem 14E: In each of the following parts, a relation is defined on the set of all human beings. Determine...
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4. For each of the following relations, determine whether the relation is: • Transitive. • A partial order. • A strict order. • An equivalence relation. Reflexive. Anti-reflexive. • Symmetric. Anti-symmetric. Justify all your answers. a. R is a relation on the set of all people such that (a, b) e R if and only if a and b have a common grandparent. b. Ris a relation on Z such that (x, y) E R if and only if |x – y| < 2. c. Ris a relation on Z* such that (x, y) E R if and only if x is divisible by y. Hint: An integer x is divisible by an integer y with y # 0 if and only if there exists an integer k such that x = yk. d. Ris a relation on Z* such that (x, y) E R if and only if there is a positive integer n such that x" = y. R is a relation on Z x Z such that ((a, b), (c, d)) e R if and only if a <c and b < d. е.
Transcribed Image Text:4. For each of the following relations, determine whether the relation is: • Transitive. • A partial order. • A strict order. • An equivalence relation. Reflexive. Anti-reflexive. • Symmetric. Anti-symmetric. Justify all your answers. a. R is a relation on the set of all people such that (a, b) e R if and only if a and b have a common grandparent. b. Ris a relation on Z such that (x, y) E R if and only if |x – y| < 2. c. Ris a relation on Z* such that (x, y) E R if and only if x is divisible by y. Hint: An integer x is divisible by an integer y with y # 0 if and only if there exists an integer k such that x = yk. d. Ris a relation on Z* such that (x, y) E R if and only if there is a positive integer n such that x" = y. R is a relation on Z x Z such that ((a, b), (c, d)) e R if and only if a <c and b < d. е.
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