Decoding Equivalence Relations

Question

Working on a problem set in which I need to evaluate a set of relations for transitiveness, symmetry, and reflexiveness. Intuitive relations seem graspable but more difficult relations just seem untouchable to me and I don't even know where to begin. In short, I'm in need of some help learning how to decode more complex relations.

A brief and simple example to make sure I'm going about this the right way:

R= To be adjacent

Reflexive (a~a): No -- X cannot be adjacent of itself

Symmetric (a~b): Yes -- X can be adjacent to Y

Transitive (a~b,b~c, therefore a~c): No -- If X is adjacent left to Y and Y is adjacent left of Z, X and Z are not adjacent

More problematic relations:

Edit: I've noticed the intuitive problem here for me is that these relations aren't simple predicates like taller, loves, bigger. Instead, the relation itself seems more like a~b from the get-go contingent on two things meeting certain criteria.

1. the relation defined on pairs of integers that holds just in case both integers are even.

In this case, my confusion lies in whether each variable is a pair of integers or a single integer. If it is a pair how do we even test these for symmetry or transitiveness? If its a single integer how can it possibly have the relation of being a pair of two even integers?

2. the relation defined on natural numbers of having a common factor greater than 3

Again whats our domain? If it's all natural numbers any number that doesnt have 3 as a factor is a counter-example of reflexiveness. If it is just numbers with 3 as a factor then how do we test for symmetry? How are we to relate two numbers a~b when the relation itself is a a~b (natural numbers who have a common factor greater than 3) relation.

Answer 1

I think you have to assume that these relations are being defined on all integers. But it should say that somewhere 1)

a~b if a and b are both even numbers

Symmetry a~a -- not if a is an odd number Reflexive and transitive hold

2) a~b if $\gcd(a,b) > 3$ Symmetry not if $|a|<3$

Reflexive holds.

Transitivity

let $a = 5, b = 20, c = 4$

a ~ b, b~c but not a~c