   Chapter 1.7, Problem 23E

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# A relation R on a nonempty set A is called antisymmetric if, for x and y in A , x R y and y R x together imply x = y . (That is, R is antisymmetric if x ≠ y implies that either x R y   or y R x .) Which of the relations in Exercise 2 areasymmetric?In each of the following parts, a relation R is defined on the set ℤ of all integers. Determine ineach 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 < 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 | < 1 .

(a)

To determine

Whether the relation R defined by xRy if and only if x=2y is antisymmetric or not.

Explanation

Given information:

The relation R defined on set of all integers by xRy if and only if x=2y.

A relation R on non-empty set A is called antisymmetric if, for x,yA xRy and yRx implies that x=y.

Explanation:

Let x,y.

xRyx=2y and yRxy=2x</

(b)

To determine

Whether the relation R defined by xRy if and only if x=y is antisymmetric or not.

(c)

To determine

Whether the relation R defined by xRy if and only if y=xk for some k is antisymmetric or not.

(d)

To determine

Whether the relation R defined by xRy if and only if x<y is antisymmetric or not.

(e)

To determine

Whether the relation R defined by xRy if and only if xy is antisymmetric or not.

(f)

To determine

Whether the relation R defined by xRy if and only if x=|y| is antisymmetric or not.

(g)

To determine

Whether the relation R defined by xRy if and only if |x||y+1| is antisymmetric or not.

(h)

To determine

Whether the relation R defined by xRy if and only if xy0 is antisymmetric or not.

(i)

To determine

Whether the relation R defined by xRy if and only if xy0 is antisymmetric or not.

(j)

To determine

Whether the relation R defined by xRy if and only if |xy|=1 is antisymmetric or not.

(k)

To determine

Whether the relation R defined by xRy if and only if |xy|<1 is antisymmetric or not.

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