Question

Let R be a relation on the set A of ordered pairs of positive integers defined by `(x , y) R (u , v)`if and only if `x v = y u`. Show that R is an equivalence relation.

Answer 1

If `(x,y)R(u,v)`, then `xv = yu`
If `(x,y)R(x,y)`, then `xy = yx`
As `xy = yx`
`:. R` is reflexive.

If `(x,y)R(u,v)`, then `xv = yu`
Now, if `(u,v)R(x,y)`, then `uy = vx`
As `xv = yu => vx = yu =>uy = vx`
`:. R` is symmetric.
If `(x,y)R(u,v)`, then `xv = yu`->(1)
If `(u,v)R(a,b)`, then `ub = va`
As `u = (va/b)` Putting value of `u` in (1),
`xv = y(va)/b=>xvb = yva =>xb = ya`
Hence, `(x,y)R(a,b)`
`:. R` is transitive.
As `R` is reflexive, summetric and transitive, `R` is an equivalence relation.