   Chapter 1.7, Problem 2E

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# 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 < 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 ≥ 0 .i. x R y if and only if x y ≤ 0 .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, “xRy if and only if x=2y” defined on set Z of all integers is reflexive, symmetric or transitive or not.

Explanation

For relation R to be reflexive for every xZ we must have xRx, but it is not true in case of x=2y because x=2x is true if and only if x=0.

Therefore, R is not reflexive.

For relation R to be symmetric, we have that xRy implies yRx,

x=2y y=2x

which is not true

(b)

To determine

Whether the relation, “xRy if and only if x=y” defined on set Z of all integers is reflexive, symmetric or transitive or not.

(c)

To determine

Whether the relation, “xRy if and only if y=xk for some k in Z” defined on set Z of all integers is reflexive, symmetric or transitive or not.

(d)

To determine

Whether the relation, “xRy if and only if x<y” defined on set Z of all integers is reflexive, symmetric or transitive or not.

(e)

To determine

Whether the relation, “xRy if and only if xy” defined on set Z of all integers is reflexive, symmetric or transitive or not.

(f)

To determine

Whether the relation, “xRy if and only if x=|y|” defined on set Z of all integers is reflexive, symmetric or transitive or not.

(g)

To determine

Whether the relation, “xRy if and only if |x||y+1|” defined on set Z of all integers is reflexive, symmetric or transitive or not.

(h)

To determine

Whether the relation, “xRy if and only if xy0” defined on set Z of all integers is reflexive, symmetric or transitive or not.

(i)

To determine

Whether the relation, “xRy if and only if xy0” defined on set Z of all integers is reflexive, symmetric or transitive or not.

(j)

To determine

Whether the relation, “xRy if and only if |xy|=1” defined on set Z of all integers is reflexive, symmetric or transitive or not.

(k)

To determine

Whether the relation, “xRy if and only if |xy|<1” defined on set Z of all integers is reflexive, symmetric or transitive or not.

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