Home
Class 12
MATHS
N is the set of natural numbers. The rel...

N is the set of natural numbers. The relation R is defined on `NxxN` as follows
`(a,b)R(c,d)iffa+d=b+c`
Prove that R is an equivalence relation.

A

Both S and T are equivalence relations on R

B

S is an equivalence relation on R but T is not

C

T is an equivalence relation on R but S is not

D

Neither S nor T is an equivalence relations on R

Text Solution

Verified by Experts

The correct Answer is:
C
`T={(x,y):x-yinI}`
As `0 in I`, so T is a reflexive relation
If `x-yinIimpliesy-x inI`
`therefore` T is symmetric also.
If x - y = I and y - z = 12
Then, `x-z=(x-y)+(y-z)=I_(1)+I_(2)inI`
`therefore` T is also transitive.
Hence, T is an equivalence relation. Clearly, `xnex+1implies(x,x)cancelinS`
`thereforeS` is not reflexive.
Promotional Banner

Topper's Solved these Questions

  • SETS, RELATIONS AND FUNCTIONS

    ARIHANT MATHS|Exercise Exercise (Subjective Type Questions)|15 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 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

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.

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.

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 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 A = (1,2,3,……….,9) and R be the relation in AxxA defined by (a,b) R (c,d) if: a+d=b+c for (a,b), (c,d) in AxxA . Prove that R is an equivalence relation. Also obtain the equivalence class {(2,5)}

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

On the set N of all natural numbers, a relation R is defined as follows: AA n,m in N , n R m Each of the natural numbers n and m leaves the remainder less than 5.Show that R is an equivalence relation. Also, obtain the pairwise disjoint subsets determined by R.

Prove that the relation R defined on the set Z of integers as R= {(a, b) : 4 divides abs(a-b)} is an equivalence relation.

Prove that the relation R defined on the set Z of integers as R= {(a, b) : 5 divides abs(a-b)} is an equivalence relation.