Question

a relation R is defined in NxN such that (a,b)R(c,d) if a+d=b+c .prove that relation R is equivalence relation.

Answer 1

A binary relation is an equivalence relation iff it satisfies the following properties:
1. reflexivity
2. symmetry
3. transistivity.
using the condition a+d=b+c, are you able to show that the given relation satisfies all of the above properties?
@heena94637

Answer 2

I know about these properties but I am confused how to prove these properties in this equation .please explain if u can.

Answer 3

Hints:
Given (a,b)R(c,d) where a+d=b+c.
1. Reflexivity : operating on itself
you need to show (a,b)R(a,b) satisfies the relation:

substitute a=a, b=b, c=a, d=b into a+d=b+c to get a+b=b+a which is true.
Therefore reflexivity is satisfied.
2. Symmetry : switching the first and second operands
show that (c,d)R(a,b) is a relation R.
3. Transistivity: show that if (a,b)R(c,d), (c,d)R(e,f) are both relations, then (a,b)R(e,f) is also a relation.