Question

The following relations are defined on the set of real numbers.
(i) `a R b iff |a - b| gt 0`
(ii) `a R b iff |a| = |b|`
(iii) `a R b iff |a| ge |b|`
(iv) `a R b iff 1 + ab gt 0`
(v) `a R b iff |a| le b`
Find whether these relations are reflexive, symmetric or transitive.

Answer 1

Correct Answer - (i) Not reflexive, symmetric, not transitive
(ii) Reflexive, symmetric, transitive
(iii) Reflexive, symmetric, not transitive
(v) Not reflexive, not symmetric, transitive
(i) `aRb iff |a-b|gt0`
Reflexivity a - a = 0
`therefore (a, a) cancelinR`
`therefore` R is not reflexive
Symmetry `(a, b) in Rimplies|a-b|gt0`
`implies |-(b-a)gt0|`
`implies |b-a|gt0`
`implies (b, a) in R`
`therefore` R is symmetric relation.
Transitivity `(a, b) in R and (b, c) in R`
`implies |a-b|gt0and|b-c|gt0`
`implies |a-b|+|b-c|gt0` [by addition]
Now, let `a gt b` and `b gt c`, then `a gt c`
`|a-b|+|b-c|=a-b+b-c=a-cgt0`
implies `|a-c|gt0`
If `a lt b` and `b gt c`, then
`|a-b|+|b-c|=-(a-b)+(b-c)=-a+2b-c`
`cancelimplies |a-c|gt0`
`therefore ` R is not transitivity relation.
(ii) `aRbiff|a|=|b|`
Reflexivity We have, |a| = |a|
`implies aRaAAa`
`therefore` R is symmetric relation.
Transitivity `(a, b) in R and (b, c) in R`
implies |a| = |b| and |b| = |c|
implies |a| = |c|
implies `(a, c) in R`
`therefore R` is transitive relation.
(iii) `aRb iff |a| ge |b|`
Reflexivity For any `a in R`, we have `|a| ge |a|`
So, `aRa AA a`
`therefore` R is not symmetric relation.
Symmetry `aRb implies |a| ge |b|`
implies `|b| le |a|`
`therefore` R is not symmetric relation.
Transitivity aRb and bRc implies |a| `ge` |b| and `|b| ge |c|`
implies `|a| ge |c|`
implies aRc
`therefore` R is reflexive on R.
Symmetry Let `(a, b) in R`, then (a, b) `in` R
implies `1 + ab gt 0`
implies 1+ ba `gt` 0
implies (b, a) `in` R
`therefore` R is symmetric on R.
Transitivity We observe that `(1, (1)/(2)) in R` and `((1)/(2), -1) in R` but `(1, -1) cancelin R` because
1 + (1) (-1) = 0 `cancelgt` 0
`therefore` R is not transitive on R.
(v) `aRb iff |a| le b`
Reflexivity Let -1 `in R`, then `|-1|cancelle(-1)`
`therefore R` is not reflexive relation
Symmetry Now, let - 3 R 4, then `|4| cancelle -3`
`therefore` R is not symmetric relation
Transitivity aRb and bRc implies `|a| le b and |b| le c`
Then, `|a| le c implies aRc`
`therefore R` is transitive relation.