Reflexive, symmetric, transitive sets? I am struggling to understand how to solve this task. 
Consider a relation R defined on the set of integers. Determine for the following if the relations are reflexive, symmetric, transitive.
a. $R =\{(,)|=3\}$
b. $R =\{(,)|+\geq0\}$
c. $R=\{(,)|^2>\}$
d. $R =\{(,)|−3>\}$
In example a, am I supposed to consider a,b and 3 as part of the set?
If so, how is this solved with "-" and ">" etc?
 A: I will solve part (a) only to give you an idea about how you can solve problems like this.
$$R=\{(a,b) \in \mathbb{Z}\times \mathbb{Z} \mid a=3b \}$$
Is $R$ reflexive? To be reflexive, we must have that for any integer $a\in \mathbb{Z}$, $(a,a) \in R$. But $(a,a)\in R$ means that $a=3a$ which is false for all integers except $a=0$. So, the relation is not reflexive.
Is it symmetric? To be symmetric, we must have that $(a,b) \in R \implies (b,a) \in R$. Does $a=3b$ imply $b=3a$? Well, no. Because $a=3b$ and $b=3a$ result in the equation $a=9a$ which is again true for $a=0$ only. So, only the pair $(0,0)$ satisfies the condition for symmetry while if $R$ were symmetric, all pairs had to satisfy this condition. So, $R$ is not a symmetric relation.
Is it transitive? Well, transitivity means that $(a,b) \in R$ and $(b,c) \in R$ should imply $(a,c) \in R$.
$a=3b$ and $b=3c$ imply $a=9c$ instead of $a=3c$. So, the relation is not transitive either.
Now try to check reflexivity, symmetry and transitivity for other relations given in the question.
A: How to read these:
When describing a set like $R = \{(a, b)\mid a = 3b\}$, this is called set builder notation. It's a common way to write a set by describing all its elements instead of having to list them all.
Set builder notation works like this: $\{x \mid \varphi(x)\}$ denotes the set of all $x$ which fulfill the condition $\varphi(x)$. In the first example, you have $(a, b)$ as the first term here, and we're told that $R$ is a relation on the integers, so that means that $R$ consists of all integer pairs $(a, b)$ such that [something]. This something is written as $a = 3b$, and as $a$ and $b$ are integers, this is really read straight-forward an means exactly what it looks like it means.
So, the first relation is defined as "The set of all pairs of integers such that the first element is equal to three times the second element." So $(3, 1)$ and $(30, 10)$ and $(-6, -2)$ are in $R$, but $(1, 2)$ and $(5, -8)$ are not.
