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)].
A. symmetric only
B. symmetric and transitive only
C. equivalence relation
D. Reflexive only

Answer 1

Correct Answer - C
(a, b) R (c, d) `iff` a + d = b + c
(i) a+a=a+a.
`therefore (a, a)R(a, a)` implies R is reflexive.
(ii) `(a, b) R (c, d)impliesa+d=b+c`
`(c, d) R (a, b) implies c+b=d+a`
`therefore` R is symmetric.
(iii) Let (a, b) R (c, d) and (c, d) R (e, f)
`impliesa+d=b+c and c+f=d+e`
`impliesa+d+c+f=b+c+d+e`
`impliesa+f=b+e`
`implies(a,b)R(e,f)`
`therefore` R is transitive.
from (i), (ii), (iii) R is an equivalence relation.