4. (8 points) For both of the following relations on the integers, give 3 pairs

Question

4. (8 points) 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). Note that the first relation is on the set of natural numbers, and the second is on the set of real numbers (a) R3 (i ,j) i, i E N, and i j 1 Three pairs of elements of N related under R3 are: (1,1), (123,7), (12, 3) Any pair of natural numbers where the first is not less than the second. and Three pairs of elements of R related under R4 are (T, 3.7), 2, 1.9), (-4, -3.6 Any pair of real numbers that do not have an integer between them

Explanation / Answer

Part (a) : For a relation to be reflexive, one ordered pair of its set must have both elements same and here we find one such pair (1,1) that also fulfill the condition that a/b=1, 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/b>=1 if (a,b) is in R then (b,a) should also be in R but this is also not being following in given set as if 12/3>1 then 3/12 is not greater than 1 , 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 but here in given three orderd pairs, this is not followed, so this relation is not reflexive. for example |pi|=|3.7| that is not true

Further by rule, a relation R is symmetric, if for any two integer numbers a and b,such that |a|=|y| 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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