Home
Class 12
MATHS
R(1) on Z defined by (a,b)inR(1) " iff "...

`R_(1)` on Z defined by `(a,b)inR_(1) " iff "|a-b|le7, R_(2)` on Q defined by `(a,b)inR_(2) " iff "ab=4 and R_(3)` on R defined by `(a, b)inR_(3)" iff "a^(2)-4ab+3ab^(2)=0`
Relation `R_(2)` is

A

reflexive

B

symmetric

C

transitive

D

equivalence

Text Solution

Verified by Experts

The correct Answer is:
A
We have, (a, b) `in R_(3)` iff `a^(2) - 4ab + 3b^(2) = 0`
where a, b `in R`
Reflexivity
`therefore a^(2)-4a.a+3d^(2)=4a^(2)-4a^(2)=0`
`therefore (a, a) in R_(3)`
`therefore` The relation `R_(3)` is reflexive.
Symmetry
`(a, b) in R_(3)`
`implies a^(2)-4ab+3b^(2)=0`, we get a = b and a = 3b
and `(b, a) in R_(3)`
implies `b^(2) - 4ab + 3a^(2) = 0`
we get b = a and b = 3a
`therefore (a, b) in R_(3) cancelimplies(b, a)in R_(3)`
`therefore` The relation `R_(3)` is not symmetric.
Transitivity We have `(3, 1), (1, (1)/(3))inR_(3)`
because `(3)^(2)-4(3)(1)+3(1)^(2)=9-12+3=0`
and `(1)^(2)-4(1)((1)/(3))+3((1)/(3))^(2)=1-(4)/(3)+(1)/(3)=0`
Also, `(3, (1)/(3))cancelinR_(3)`, because
`(3)^(2)-4.(3)((1)/(3))+3((1)/(3))^(2)=9-4+(1)/(3)=(16)/(3)ne0`
`therefore` The relation `R_(3)` is not transitive.
Promotional Banner

Topper's Solved these Questions

  • SETS, RELATIONS AND FUNCTIONS

    ARIHANT MATHS ENGLISH|Exercise Exercise (Single Integer Answer Type Questions)|5 Videos

  • SETS, RELATIONS AND FUNCTIONS

    ARIHANT MATHS ENGLISH|Exercise Exercise (Matching Type Questions)|2 Videos

  • SETS, RELATIONS AND FUNCTIONS

    ARIHANT MATHS ENGLISH|Exercise Exercise (More Than One Correct Option Type Questions)|3 Videos

  • SEQUENCES AND SERIES

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

  • THE STRAIGHT LINES

    ARIHANT MATHS ENGLISH|Exercise Exercise (Questions Asked In Previous 13 Years Exam)|17 Videos

Similar Questions

Explore conceptually related problems

Test whether, R_3 on R defined by (a ,\ b) in R_3 a^2-4\ a b+3b^2=0 .

If R is a relation on NxxN defined by (a,b) R (c,d) iff a+d=b+c, then

Test whether the following relations R_1,R_2"and"R_3,"" are (i) reflexive (ii) symmetric and (iii) transitive: R_1 on Q_0 defined by (a , b)R_1 a=1/b R_2 on Z defined by (a , b)R_2|a-b|lt=5 R_3 on R defined by (a , b)R_3 a^2-4a b+3b^2=0

Let R_(1) be a relation defined by R_(1)={(a,b)|agtb,a,b in R} . Then R_(1) , is

Let relation R defined on set of natural number is given by R = {(a,b) : a^(2)+b^(2)-4bsqrt(3)-2a sqrt(3)+15=0} then relation R is

Let R_(1) be a relation defined by R_(1)={(ab)|ageb,a,bepsilonR} . Then R_(1) is

Let the relation R be defined in N by aRb , if 2a + 3b = 30 . Then R = …… .

Let the relation R be defined in N by aRb, if 2a + 3b = 30. Then R = …… .

For each binary operation * defined below, determine whether * is commutative or associative. (i) On Z , define a ∗ b = a – b (ii) On Q , define a ∗ b = ab + 1 (iii) On Q , define a ∗ b = (ab)/(2) (iv) On Z^+ , define a ∗ b = 2^(ab) (v) On Z^+ , define a ∗ b = a^(b) (vi) On R – {– 1} , define a ∗ b = (a)/(b+1)

If R is a relation on R (set of all real numbers) defined by a R b iff ageb , then R is