Let space R space be space a space relation space f rom space N space t o space N space and space m space b e space a space given space fixed space p ositive space integer space such space that R space equals open curly brackets space left parenthesis space a comma b right parenthesis space : space a comma b space element of space Z space a n d space a minus b space i s space d i v s i b l e space b y space m space open space close curly brackets close.  S h o w space t h a t space R space e q u i v a l e n c e space r e l a t i o n space

Asked by inderjeet.saluja | 27th Aug, 2014, 12:27: AM

Expert Answer:

The Relation R is an equivalence relation if and only if it satisfies the properties, Reflexivity, Symmetry and Transitivity.
 
Reflexivity: a is related to a because a-a is zero and zero divisible by any number and hence R is reflexive.
 
Symmetry: Assume that a is related to b, and hence a-b is divisible by m
 
Since a and b are integers, we have, a minus b equals m k subscript 1 space a n d space b minus a equals m k subscript 2 comma space w h e r e space k subscript 1 space a n d space k subscript 2 space a r e space a n y space i n t e g e r s and hence both are divisible by m.
Thus, R is Symmetric.
 
Transitivity: Iif a is related to b, we have, a minus b equals m k subscript 1
If b is related to c, we have, b minus c equals m k subscript 2
Now consider,
 a minus c equals a minus b plus b minus c equals m k subscript 1 plus m k subscript 2 equals m open parentheses k subscript 1 plus k subscript 2 close parentheses
Thus, a - c is divisible by m and hence a is related to c.
Thus, R is transitive.
Hence R is an equivalence relation.

Answered by Vimala Ramamurthy | 27th Aug, 2014, 08:59: AM