Ds Assignment solution by shabi :)

Ds assignment No 3

Let R be the following relation:

(a,b) R (c,d) if and only if (a<c or a=c and b<d or b=d)

(a) Is R is reflexive?prove it or disprove it

(b) Is R is symmetric? Prove it or disprove it

(c) Is R is transitive? Prove it or disprove it

Sol:

     (a)    According to the definition of reflexivity that is :Let R be a binary relation on A.

 • R is reflexive if for all x e A, (x,x) e  R. (Equivalently, for all x e A, x R x.)

So by given condition that is

a<c or A=C  and B<D or B=D which contradicts the reflexive property

Hence it is not reflexive

      (b)    According to the definition of  symmetry that is :

• R is symmetric if for all (x,y) e A, (x,y) e R implies (y,x) e R. (Equivalently, for all (x,y) e A, x R y implies that y R x.)

So by the give condition that is

A<C or A=C and B<D or B=D which contradicts the property of symmetry

Hence it is not symmetric 

       (c)    According to the the definition of transitive that is :

• R is transitive if for all (x,y,z) e A, (x,y) e R and (y,z) e R implies (x,z) e R. (Equivalently, for all x,y,z e A, x R y and y R z implies x R z.)

So by the given condition that is

A<C or A=C and B<D or B=D  which follows the transitive property  

As (a,b) R (c,d) so aRc and aRd also bRc and bRd  as a<=c then b<c  if C<=d then also (a,b)<=d  

Hence it is transitive