On the set $\mathbb N\times \mathbb N$ define $(m,n)\sim(k,l)$ if $m+l=n+k$
- Show that $\sim$ is equivalence relation on $\mathbb N\times \mathbb N$
- Draw a sketch of $\mathbb N\times \mathbb N$ that shows several equivalence classes.
My book does not even explain how to do this kind of problems. I understand that we need to show that it's symmetric, reflexive, and transitive. However, I usually do it with a matrix. How can I put the above problem into matrix form and then draw diagrams from it?
Best Answer
(a)R in N will be reflexive iff (m,n) R (m,n).
Now, m+n=n+m$\implies$(m,n) R (m,n)$\implies$R is reflexive
(b)R in N will be symmetric if (m,n) R (l,k)$\implies$ (l,k) R (m,n). Now, m+k=n+l$\implies$l+n=k+m$\implies$ (m,n) R (l,k)$\implies$ (l,k) R (m,n)$\implies$R is symmetric.
(c)R in N will be transitive if (m,n) R (l,k) and (l,k) R (p,q)$\implies$(m,n) R (p,q). Now,
(m,n) R (l,k) and (l,k) R (p,q) $\implies$ m+k=n+l and l+q=k+p$\implies$k=n+l-m and k=l+q-p
$\implies$n+l-m=l+q-p$\implies$n-m=q-p$\implies$n+p=q+m$\implies$m+q=n+p$\implies$(m,n) R (p,q). Thus, R is transitive.