Equivalence Relation Properties

Question

I have to define whether the following relation is symmetric, reflexive, and transitive.

Define a relation R on Z as follows: (x, y) ∈ R if and only if x = |y|:

This is my answer so far:

Is R symmetric? No

Is R reflexive? No

Is R transitive? No

I am not exactly confident with this answer because I am not sure if we should check these properties based on x, y for all x, y ∈ R or based on x, y for all x, y ∈ Z

Answer 1

Note that $R$ is defined on $\mathbb Z$, so we assume that $x,y \in \mathbb Z$.

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You're correct that $R$ is not symmetric:

• Take $x = 3 \in \mathbb Z$ and $y = -3 \in \mathbb Z$. Notice that $3 = |-3|$ but $-3 \neq 3 = |3|$. So $(x,y) \in R$ but $(y,x) \notin R$.

You're correct that $R$ is not reflexive:

• Take $x = -3 \in \mathbb Z$. Notice that $-3 \neq 3 = |-3|$. So $(x,x) \notin R$.

However, $R$ is in fact transitive:

• Choose any $x,y,z \in \mathbb Z$ such that $(x,y),(y,z) \in R$ so that $x = |y|$ and $y = |z|$. Then by substituting, notice that: $$ x = |y| = ||z|| = |z| $$ So $(x,z) \in R$, as desired.