Problem understanding relations syntax.

[ktri]

I have this question:2. Let R be a relation on Z with \(\displaystyle R = {(a,b) : |a−b| < 3}.\)
(1) Is R reflexive? (If yes, prove it; if no, give a counterexample)
(2) Is R symmetric? (If yes, prove it; if no, give a counterexample)
(3) Is R antisymmetric? (If yes, prove it; if no, give a counterexample)
(4) Is R transitive? (If yes, prove it; if no, give a counterexample)
(5) Is R an equivalence relation?
(6) Is R a partial ordering?

My main issue is I'm not sure how to check if R is symetric or transitive etc. I know what those words mean:

symetric example: \(\displaystyle 4 * 5 = 5 * 4\)
transitive example: \(\displaystyle 2 < 3\) and \(\displaystyle 3 < 4\) so \(\displaystyle 2 < 4\)

but I'm not sure how to determine if R is any of those traits. Like to test if R is reflexive am I checking if
\(\displaystyle |a - b| < 3\) and \(\displaystyle 3 < |a - b|\) ? That dosen't seem right to me. I'm really just not sure what I'm comparing to what.

 

[Opalg]

I have this question:2. Let R be a relation on Z with \(\displaystyle R = {(a,b) : |a−b| < 3}.\)
(1) Is R reflexive? (If yes, prove it; if no, give a counterexample)
(2) Is R symmetric? (If yes, prove it; if no, give a counterexample)
(3) Is R antisymmetric? (If yes, prove it; if no, give a counterexample)
(4) Is R transitive? (If yes, prove it; if no, give a counterexample)
(5) Is R an equivalence relation?
(6) Is R a partial ordering?

My main issue is I'm not sure how to check if R is symetric or transitive etc. I know what those words mean:

symetric example: \(\displaystyle 4 * 5 = 5 * 4\)
transitive example: \(\displaystyle 2 < 3\) and \(\displaystyle 3 < 4\) so \(\displaystyle 2 < 4\)

but I'm not sure how to determine if R is any of those traits. Like to test if R is reflexive am I checking if
\(\displaystyle |a - b| < 3\) and \(\displaystyle 3 < |a - b|\) ? That dosen't seem right to me. I'm really just not sure what I'm comparing to what.

Hi ktri, and welcome to MHB!

Reflexive means that each element is related to itself. In the case of this relation R, you have to say whether $|a-a|<3$ (for every integer $a$).

To test R for symmetry, you have to decide whether $|a-b|<3$ implies that $|b-a|<3$.

To test R for transitivity, you have to decide whether $|a-b|<3$ and $|b-c|<3$ implies that $|a-c|<3$.