A relation R on a nonempty set A is called irreflexive if x R x for all x ∈ A . Which of the relations in Exercise 2 are irreflexive? In each of the following parts, a relation R is defined on the set ℤ of all integers. Determine in each case whether or not R is reflexive, symmetric, or transitive. Justify your answers. a. x R y if and only if x = 2 y . b. x R y if and only if x = − y . c. x R y if and only if y = x k for some k in ℤ . d. x R y if and only if x &lt; y . e . x R y if and only if x ≥ y . f. x R y if and only if x = | y | . g. x R y if and only if | x | ≤ | y + 1 | . h. x R y if and only if x y ≥ i. x R y if and only if x y ≤ j. x R y if and only if | x − y | = 1 . k. x R y if and only if | x − y | &lt; 1 .

Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
Publisher: Cengage Learning,
ISBN: 9781285463230

Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
Publisher: Cengage Learning,
ISBN: 9781285463230

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Chapter 1.7, Problem 21E
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