Let $\rho$ be the relation on $\mathbb{Z}\times\mathbb{Z}$ in which $(a,b)\rho(x,y) \iff a-b=x-y$. Prove it is an equivalence relation

Question

So I'm stuck on this question (not really sure how to approach it), I understand what an equivalence relation is (reflexive, symmetric & transitive relation at the same time), but I'm not sure how to prove it. Would be appreciated if someone can help me with solving this so I can carry this through to the next questions. Thanks heaps, any help would greatly be appreciated.

Question

Let $\rho$ be the relation on $\mathbb{Z}\times\mathbb{Z}$ in which $(a,b)\rho(x,y) \iff a-b=x-y$. Prove that $\rho$ is an equivalence relation.

Answer 1

You really just check the three properties one by one.

1. Reflexive means $(a,b)\rho(a,b)$

2. Symmetric means $(a,b)\rho(x,y) \Rightarrow (x,y)\rho(a,b)$

3. Transitive means $(a,b)\rho(x,y)$ and $(x,y)\rho(c,d)$ $\Rightarrow (a,b)\rho(c,d)$.

For each point, you will need a very simple argument. I will write down the first one for you, which is the most trivial one, and then try to figure out the rest yourself:

1. $\rho$ is reflexive: Let $(a,b) \in \mathbb{Z}\times\mathbb{Z}$, then $a-b=a-b$ (since $=$ is an equivalence relation), so $(a,b)\rho(a,b)$.

Answer 2

A more general approach.

Let $f:X\to Y$ be a function and define relation $\rho$ on $X$ by: $$u\rho v\iff f(u)=f(v)\tag1$$

It is quite obvious that:

• $\forall u\in X[f(u)=f(u)]$
• $\forall u,v\in X[f(u)=f(v)\implies f(v)=f(u)]$
• $\forall u,v,w\in X[f(u)=f(v)\wedge f(v)=f(w)\implies f(u)=f(w)]$

Equivalent statements are:

• $\forall u\in X[u\rho u]$ i.e. the relation $\rho$ is reflective.
• $\forall u,v\in X[u\rho v\implies v\rho u]$ i.e. the relation $\rho$ is symmetric.
• $\forall u,v,w\in X[u\rho v\wedge v\rho w\implies u\rho w]$ i.e. the relation $\rho$ is transitive.

So evidently a relation defined as in $(1)$ turns out to be an equivalence relation.

Now apply this principle on function $f:\mathbb Z\times\mathbb Z\to\mathbb Z$ prescribed by $\langle a,b\rangle\mapsto a-b$.