Question

Prove that the relation `R` on the set `NxxN` defined by `(a , b)R (c , d) a+d=b+c` for all `(a , b), (c , d) in NxxN` is an equivalence relation. Also, find the equivalence classes [(2, 3)] and [(1, 3)].

Answer 1

Here, `(a,b)R(c,d) <=> a+d = b+c` for all `(a,b),(c,d) in N xx N.` First we will check `R` for reflexive.
For, `(a,b)R(a,b) `,
`=>a+b = b+a`, which is true.
So, `R` is reflexive.
Now, we will check for symmetric.
For, `(a,b)R(c,d)`,
`=>a+d = b+c`
`=>b+c = a+d`
`=>c+b = a+d`
`=>(c,d) R(a,b) ` is tur.
So, `R` is symmetric.
Now, we will check `R` fo transtivity.
For, `(a,b)R(c,d) and (c,d)R(e,f)`
`=> a+d = b+c and c+f = d+e`
`=> a-b = c-d and c -d = e- f`
`=> a-b = e-f`
`=> a+f = b+e`
So, `(a,b)R(e,f)` is true.
`:. R` is transitive.
As `R` is reflexive, symmetric and transitive, `R` is an equivalence relation.