Question

Let R be a relation on the set A of ordered pairs of non-zero integers defined by (x, y) R (u, v) if xv = yu. Show that R is an equivalence relation.

Answer 1

(x, y) R (u, v) ⇔ xv = yu

Proof :

To prove that relation is equivalence, we need to prove that it is reflexive, symmetric and transitive.

Reflexivity : For Reflexivity, we need to prove that-

(a, a) ∈ R

∵ xy = yu

∴ (x, y) R (x, y)

Symmetric : For Symmetric, we need to prove that-

If (a, b) ∈ R, then (b, a) ∈ R

Let (x, y) R (u, v)

TPT (u, v) R (x, y)

Given xv = yu

⇒ yu = xv

⇒ uy = vx

∴ (u, v) R (x, y)

Transitive : : For Transitivity, we need to prove that-

If (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R

Let (x, y) R (u, v) and (u, v) R (p, q) …(i)

TPT (x, y) R (p, q)

TPT (xq = yp

From (1) xv = yu & uq = vp

xvuq = yuvp

xq = yp

∴ R is transitive

Since R is reflexive, symmetric & transitive

⇒ R is an equivalence relation.