Consider the relation R defined by xRy iff xy

Consider the relation R defined by xRy iff xy > 0 on the set R of all real numbers. (a) Is R reflexive? If so, prove it. Otherwise, provide a counterexample. (b) Is R symmetric? If so, prove it. Otherwise, provide a counterexample. (c) Is R antisymmetric? If so, prove it. Otherwise, provide a counterexample. (d) Is R transitive? If so, prove it. Otherwise, provide a counterexample.

Transcript

00:01 In this problem we are provided with the relation r equals to the set of all ordered pairs x y such that the product of x and y is greater than 0 where x as well as y belong to the set of all real numbers.

00:16 In the first subpart we are asked to check if the relation is reflexive.

00:22 So in order for a relation to be reflexive, x comma x must belong to the relation.

00:30 So in this case, x comma x will belong to the relation if x times x which is x squared is greater than 0.

00:38 It is clear that x squared is always greater than 0 except for the case when x equals to 0.

00:45 In that case we have 0 times 0 which is equal to 0 and not greater than 0.

00:50 So this implies that 0 comma 0 does not belong to the relation r which implies that the given relation is not reflexive.

01:04 So therefore this is the final answer for the first subpart.

01:09 Next, in the second subpart we are asked to check if the given relation is symmetric or not.

01:16 So in order for a relation to be symmetric, if x comma y belongs to the relation then y comma x, sorry, y comma x must also belong to the relation.

01:30 So let us assume that x comma y belongs to the relation.

01:32 The relation, this implies that xy is greater than 0 and since multiplication is commutative in real numbers, y comma x will also be greater than 0.

01:43 So this implies that y comma x belongs to the relation r.

01:48 So therefore the given relation is symmetric...

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