Mumbai University > Computer Engineering > Sem 3 > Discrete Structures

Marks: 8 Marks

Year: May 2014

Let a, b € I where I be the set of Integers.

1. R is reflexive:

   Since a-a=0=0.5, Ɏ a € I, aRa.

   Hence, R is reflexive.

2. R is symmetric:

   Suppose aRb

   a-b is divisible by 5.

   a-b = k.5 for some integer k,

   b-a = (-k).5 for some integer –k.

   bRa

Hence R is symmetric.

1. R is transitive:

   Suppose aRb and bRc

   $a - b=k_15$ and $b-c=k_25$ for some integers $k_1$ and $k_2$.

   $(a - b) + (b-c) = ( k_1+k_2)5$

   $a - c=(k_1+k_2)5$ for the integer $k_1 + k_2.$

   aRc

   Hence R is transitive.

   This proves that R is an equivalence relation on I.