How to determine the relation properties?

Question

For each of the following relations on a set X, determine which of the following properties it has: reflexive, symmetric and transitive, and explain why or why not.

 1) For X = Z, a ∼ b when a + b is odd
 2) For X = Z, a ∼ b when 3 | (a + b)
 3) For X = Z, a ∼ b when a^2 = b^2

For 1) I understand that it is not reflexive because a+a=2a and 2a is an even integer. For 2) it is not reflexive either because cannot do 3 | (even number). For 3) it is reflexive. I do not know how to determine the rest because I still can't seem to grasp the concept of symmetry and transitivity.

Answer 1

Symmetry is trivial. For example, if a + b is odd, is b + a odd?
More complicated is transitivity.
If a + b and b + c are odd, is a + c odd?

The others I leave for your enjoyment.

Answer 2

Realation is symmetric when you can swap $a$ and $b$ without changing truth value of their relation to each other, for example if line $L_1$ is parallel to line $L_2$ then it is also true that $L_2$ is parallel to $L_1$. Another example would be that if intersection of set $A$ with set $B$ is an empty set then it is also true that intersection of set $B$ with set $A$ is an empty set.

Transitivity connects $a$ and $c$ through their relation with $b$, for example if integer $a$ divides integer $b$. and $b$ divides integer $c$, then $a$ divides $c$. Also, if you can set a bijection from set $A$ onto set $B$, and from set $B$ onto set $C$ then you can set a bijection from set $A$ onto set $C$.

So, basically, in the world of relations, symmetry has some flavour of commutativity and transitivity has some flavour of cancellation (at least for me).

1) If $a+b$ is odd and $b+c$ is odd then what can you say about $a+c=(a+b)+(b+c)-2b$ ? (this is the check of transitivity).

2) If $3|(a+b)$ and $3|(b+c)$ then what can you say about divisibility by $3$ of $a+c=(a+b)+(b+c)-2b$.

If you are not sure about this one then try to find some concrete $a,b,c$ for which transitivity could fail, if possible? If it fails in one case then it does not have desired property, by definition.

3) If $a^2=b^2$ and $b^2=c^2$, is $a^2=c^2$?

I bet that you can handle symmetry now, can you?