Home
Class 12
MATHS
Let N denote the set of all natural numb...

Let `N` denote the set of all natural numbers and R be the relation on `NxN` defined by `(a , b)R(c , d) `<=>` a d(b+c)=b c(a+d)dot` Check whether R is an equivalence relation on `NxxNdot`

Text Solution

Verified by Experts

Reflexive
Since, `(a,b)R(a,b)iffab(b+a)=ba(a+b),AAa,binN` is true.
Hence, R is reflexive.
Symmetric (a, b) R (c, d)
`iff ad(b+c)=bc(a+d)`
`iff bc(a+d)=ad(b+c)`
`iff cb(d+a)=da(c+b)`
`iff (c,d)R(a,b)`
Hence, R is symmetric.
Transitive
Since, `(a,b)R(c,d)iffad(b+c)=bc(a+d)`
`iff (b+c)/(bc)=(a+d)/(ad)`
`if (1)/(c)+(1)/(b)=(1)/(d)+(1)/(a)`
`iff (1)/(a)-(1)/(b)=(1)/(c)-(1)/(d)`
`therefore (a,b)R(c,d)iff(1)/(a)-(1)/(b)=(1)/(c)-(1)/(d)" ... (i)"`
and similarly (c,d) R (e,f) `iff (1)/(c)-(1)/(d)=(1)/(e)-(1)/(f)" ... (ii)"`
From Eqs. (i) and (ii),
`(a,b)R(c,d)and(c,d)R(e,f)iff(1)/(a)-(1)/(b)=(1)/(e)-(1)/(f)iff(a,b)R(e,f)`
So, R is transitive. Hence, R is an equivalence relation.
Promotional Banner

Topper's Solved these Questions

  • SETS, RELATIONS AND FUNCTIONS

    ARIHANT MATHS|Exercise Exercise For Session 1|11 Videos

  • SETS, RELATIONS AND FUNCTIONS

    ARIHANT MATHS|Exercise Exercise For Session 2|10 Videos

  • SEQUENCES AND SERIES

    ARIHANT MATHS|Exercise Exercise (Questions Asked In Previous 13 Years Exam)|38 Videos

  • THE STRAIGHT LINES

    ARIHANT MATHS|Exercise The Straight Lines Exercise 8 : (Questions Asked in Previous 13 years Exams)|1 Videos

Similar Questions

Explore conceptually related problems

Let N be the set of natural number and R be the relation in NxxN defined by : (a,b) R (c,d) iff ad = bc, for all (a,b), (c,d) in NxxN Show that R is an equivalence relation.

Let S be the set of all real numbers and let R be the relation in S defined by R = {(a,b), a leb^2 }, then

If R is the relation in N xx N defined by (a, b) R (c,d) if and only if (a + d) =(b + c), show that R is an equivalence relation.

Let Z be the set of all integers and R be the relation on Z defined as R = (a,b) : a,b in Z and a-b is divisible by 5) Prove that R is an equivalence relation.

Let N be the set of all natural numbers and R be a relation in N defined by : R = {(a, b) : a is a factor of b} Show that R is reflexive and transitive.

If R is a relation in N xx N , show that the relation R defined by (a, b) R (c, d) if and only if ad = bc is an equivalence relation.

Show that the relation R defined by (a, b) R(c,d)implies a+d=b+c in the set N is an equivalence relation.

Let f : X rarr Y be a function. Define a relation R in X given by R = {(a, b): f (a) = f (b)} Examine whether R is an equivalence relation or not.

Show that the relation R defined by R = {(a, b) (a - b) , is divisible by 5, a, b in N} is an equivalence relation.

If relation R defined on set A is an equivalence relation, then R is