Show that the relation $\sim$ defined on the set $X = \mathbb{N} \times \mathbb{N} = \{(a, b) : a \in \mathbb{N}; b \in \mathbb{N}\}$ as
$(a,b) \sim (c,d)$ if and only if $a + d = c + b$ is an equivalence relation.
(ii) Write down three elements of the equivalence class of $(12, 7)$. What do they all have in common?
I know that for a relation to be an equivalence relation it should be reflexive, symmetric and transitive. I'm just not really sure how to apply that to the question.
For the second part, I don't fully understand the concept of what an equivalence class is or what the question means.
Any help would be appreciated.
Best Answer
For reflextive : $a+b=b+a$ and hence $(a,b)\sim (a,b)$.
symmetric : Let $(a,b)\sim (c,d)\Rightarrow\ a+d=b+c\Rightarrow\ d+a=c+b\Rightarrow (c,d)\sim (a,b)$
transitive : let $(a,b)\sim (c,d)$ and $(c,d)\sim (e,f)$, so $a+d=b+c,\ c+f=d+e\Rightarrow\ a+(c+f-e)=b+c\Rightarrow\ a+f=b+e$ and hence $(a,b)\sim (e,f)$. Since $\sim$ is reflexive, symmetric and transitive hence it's an equivalence relation.
Equivalence class of any element $(a,b)=\{(c,d)|a+d=b+c\}$, so can you find three such $(a,b)'s$ such that $12+b=7+a$.