For both of the following relations on the integers, • give 3 pairs of elements

ID: 3032605 • Letter: F

Question

For both of the following relations on the integers,

• give 3 pairs of elements that are related,

• determine whether the relation is reflexive,

• determine whether the relation is symmetric and

• determine whether the relation is transitive.

You must prove your answer for each of the three properties (reflexive, symmetric and transitive).

(a) R1 = {(a, b) : a, b Z, and a · b > 0} Three pairs of elements of Z related under R1 are: (1, 3),(123, 7),(1, 24). (Any pair of non-zero integers which are both positive or both negative.)

(b) R2 = {(n, m) : n, m Z, and n · m 0} Three pairs of elements of Z related under R2 are: (0, 0),(123, 7),(1, 24). (Any pair of integers except those with one positive and one negative integer.)

Explanation / Answer

(a) If a R b , then a and b are either both positive or both negative as a. b > 0. The relation R is apparently reflexive as for any non –zero a Z, a. a > 0 regardless of a being positive or negative.

The relation is symmetric also because if a. b > 0 , then b. a > 0 ( as a. b = b. a). Thus a R b implies that b R a.

The relation is also transitive because if a R b, then a. b > 0 so that a and b are either both positive or both negative. Similarly if b R c , then b. c> 0 so that b and c are either both positive or both negative. Considering these 2 statements together, a and c are either both positive or both negative so that a . c > 0 and hence a R c.

(b) For any a Z, since a. a 0 , hence a R a for all a and hence, the relation R is reflexive.

If a R b, then b. a 0. Since b. a = a. b, hence a. b is also 0 so that b R a. Thus the relation is symmetric.

If a, b, c Z such that b. a 0, then a, b are either both positive or both negative or either a = 0 or b = 0 or both a, b = 0. Similarly, if b. c 0, then b, c are either both positive or both negative or either b = 0 or c = 0 or both b, c = 0. Considering these 2 statements together, a and c are either both positive or both negative , or either a = 0 or c = 0 or both a, c = 0 so that a. c 0. Hence a R c so that the relation is transitive.

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For both of the following relations on the integers, • give 3 pairs of elements

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