Let R be a relation on the set A of ordered pair of integers defin

1.	Let R be a relation on the set A of ordered pair of integers defined by (x, y) R (u, v) if xv = yu. Show that R is an equivalence relation.
Answer» First let R be a relation on A Given as set A of ordered pair of integers defined by (x, y) R (u, v) if xv = yu Let us check whether the given relation is equivalence or not. To prove equivalence relation, the given relation should be reflexive, symmetric and transitive. Reflexivity: Let (a, b) be an arbitrary element of the set A. Then, (a, b) ∈ A ⇒ ab = ba ⇒ (a, b) R (a, b) So, R is reflexive on A. Symmetry: Let (x, y) and (u, v) ∈ A such that (x, y) R (u, v). Then, xv = yu ⇒ vx = uy ⇒ uy = vx ⇒ (u, v) R (x, y) Thus, R is symmetric on A. Transitivity: Let (x, y), (u, v) and (p, q) ∈ R such that (x, y) R (u, v) and (u, v) R (p, q) ⇒ xv = yu and uq = vp By, multiplying the corresponding sides, we get xv × uq = yu × vp ⇒ xq = yp ⇒ (x, y) R (p, q) So, R is transitive on A. ∴ R is reflexive, symmetric and transitive. Hence, R is an equivalence relation on A.

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

Answer»

First let R be a relation on A

Given as set A of ordered pair of integers defined by (x, y) R (u, v) if xv = yu

Let us check whether the given relation is equivalence or not.

To prove equivalence relation, the given relation should be reflexive, symmetric and transitive.

Reflexivity:

Let (a, b) be an arbitrary element of the set A.

Then, (a, b) ∈ A

⇒ ab = ba

⇒ (a, b) R (a, b)

So, R is reflexive on A.

Symmetry:

Let (x, y) and (u, v) ∈ A such that (x, y) R (u, v). Then,

xv = yu

⇒ vx = uy

⇒ uy = vx

⇒ (u, v) R (x, y)

Thus, R is symmetric on A.

Transitivity:

Let (x, y), (u, v) and (p, q) ∈ R such that (x, y) R (u, v) and (u, v) R (p, q)

⇒ xv = yu and uq = vp

By, multiplying the corresponding sides, we get

xv × uq = yu × vp

⇒ xq = yp

⇒ (x, y) R (p, q)

So, R is transitive on A.

∴ R is reflexive, symmetric and transitive.

Hence, R is an equivalence relation on A.