Home
Class 12
MATHS
Consider the following relations: R = {(...

Consider the following relations: R = {(x, y) | x, y are real numbers and x = wy for some rational number w}; `S={(m/n , p/q)"m , n , pandqa r ei n t e g e r ss u c ht h a tn ,q"!="0andq m = p n"}` . Then (1) neither R nor S is an equivalence relation (2) S is an equivalence relation but R is not an equivalence relation (3) R and S both are equivalence relations (4) R is an equivalence relation but S is not an equivalence relation

A

neither R nor S is an equivalence relation

B

S is an equivalence relation but R is not an equivalence relation

C

R and S both are equivalence relations

D

R is an equivalence relation but S is not an equivalence relation

Text Solution

Verified by Experts

The correct Answer is:
B
xRy need not implies yRx.
`S:(m)/(n)S(P)/(q)iffqm="pm"implies(m)/(s)S(m)/(n)` is reflexive. `(m)/(s)S(p)/(q)implies(p)/(q)S(m)/(n)` is symmetric.
and `(m)/(n)S(p)/(q),(p)/(q)S(r)/(t)impliesqm=pn,pt=qr`
`implies mt=nrimplies(m)/(n)S(r)/(t)` is transitive.
`therefore` S is an equivalence relation.
Promotional Banner

Similar Questions

Explore conceptually related problems

If R_1 and R_2 are equivalence relations in a set A, show that R_(1) cap R_2 is also an equivalence relation.

If R = set of real number and Q = set of rational number then R - Q = …….

The relation R difined the set Z as R = {(x,y) : x - yin Z} show that R is an equivalence relation.

Let A = {p, q, r}. Which of the following is an equivalence relation on A?

R is relation in N xxN as (a,b) R (c,d) hArr ad = bc . Show that R is an equivalence relation.

Statement-1: The relation R on the set N xx N defined by (a, b) R (c, d) iff a+d = b+c for all a, b, c, d in N is an equivalence relation. Statement-2: The intersection of two equivalence relations on a set A is an equivalence relation.

Let R be the real line. Consider the following subsets of the plane RxxR . S""=""{(x ,""y)"":""y""=""x""+""1""a n d""0""<""x""<""2},""T""=""{(x ,""y)"":""x-y"" is an integer }. Which one of the following is true? (1) neither S nor T is an equivalence relation on R (2) both S and T are equivalence relations on R (3) S is an equivalence relation on R but T is not (4) T is an equivalence relation on R but S is not

x and y are real numbers . If xRy hArr x - y +sqrt5 is on irrational number then R is ......... Relation .

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