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.

  • 2
    $\begingroup$ Do you understand the definition of an equivalence relation? $\endgroup$ – Vincent Boelens Apr 20 '14 at 14:06

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 $12+b=7+a$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.