a Define R on Z as follows For all m n Z m R n iff m n 100 i

(a) Define R on Z as follows: For all m, n Z

m R n iff |m n| 100

i. Prove that R is reflexive.

ii. Prove that R is symmetric.

iii. Prove that R is not transitive.

(b) Define a relation E on R (the reals) as follows: For all x, y R

xEy iff xy > 0 or x = y = 0

Prove that E is an equivalence relation. How many equivalence

classes are there?

Solution

a)

i.

For any integer m

|m-m|=0<=100

Hence, mRm

Hence R is reflexive

ii.

For any integers , m,n

|m-n|=|n-m|

hence, mRn implies nRm

hence R is symmetrc

iii.

Let, m=-50,n=50,p=100

Hence, mRn as |m-n|=100

nRp as |n-p|=50

But |m-p|=150

Hence R is not transitive

b)

i. For any x real, x^2>=0

Hence, xRx

Hence, R is reflexive

ii. Let, xRy

so, xy>0 or x=y=0

Hence, yRx

iii. Let, xRy,yRz

so, xy>0 or x=y=0

yz>0 or y=z=0

Case 1: xy>0

Hence yz>0

Hence x and z both have the same sign as y

Hence x and z have teh same sign

Hence, xz>0

Case 2 x=y=0

Hence, y=z=0

Hence, x=z=0

Hence, xRz

So R is an equivalence relation.

Three equvalence classes

1. Set of all positive integers

2. Set of all negative integers

3. {0}