give 3 pairs of elements that are related, determine whether the relation is ref
ID: 3012411 • Letter: G
Question
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 tran sitive). (a) Ri (a, b) a, b E Z, and a b 0 Three pairs of elements of Z related under Ri are: (1,3), (123,7), (-1 24) (Any pair of non-zero integers which are both positive or both negative.) (b) R2 m, m) n, m e 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 integerExplanation / Answer
Part (a) : For a relation to be reflexive, one ordered pair of its set must have both elements same but here in given three orderd pairs, this is not followed, so this relation is not reflexive.
Further by rule, a relation R is symmetric, if for any two integer numbers a and b,such that a x b >0 if (a,b) is in R then (b,a) should also be in R but this is also not being following in given set , so this relation is not symmetric.
Also by rule, a relation R is transitive, if for any three real numbers a,b and c, if (a,b) and (b,c) hold in R, then (a,c) must also be there in R. So here also, we find that this condition is not being ful filled, so given relation is not transitive also.
This is the answer of part (a)
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Part (b) : For a relation to be reflexive, one ordered pair of its set must have both elements same and here we find one such pair (0,0) that also fulfill the condition that a x b =0, so clearly relation R2 is reflexive.
Further by rule, a relation R is symmetric, if for any two integer numbers a and b,such that a x b >0 if (a,b) is in R then (b,a) should also be in R but this is also not being following in given set , so this relation is not symmetric.
Also by rule, a relation R is transitive, if for any three real numbers a,b and c, if (a,b) and (b,c) hold in R, then (a,c) must also be there in R. So here also, we find that this condition is not being ful filled, so given relation is not transitive also.
This is the answer of part (b)
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