equivalence relation proof for all positive integers

Question

The relation $R$ is defined for all positive integers such that $(a,b) R (c,d) \longleftrightarrow a+d=b+c$. Show that $R$ is an equivalence relation.

Answer 1

For R to be an equivalence relationship it would have to suffice all the following conditions :

1) it has to be a reflexive relationship ! ( (a,b) R (a,b) )

2) it has to be simetric ( (a,b) R (c,d) only if (c,d) R (a,b) )

3) it has to be transitive ( if ( a,b) R (c,d) and (c,d) R (e,f) then (a,b) R (e,f) ) ;

Hope this was helpful!

Answer 2

Note that $\left( a,b\right) R \left( c,d\right)$ if and only if $a-b = c-d$.