Let P(S) denote the power set of S = { 1,2,3,...,10}.

Define the relations R_{1} and R_{2} on P(S) as

AR_{1}B if

( A cap B^{C} ) cup ( B cap A^{C} ) = phi

and AR_{2}B if cup B^{C} = B cup

A^{C},forall A,B in P(S). Then

Question

Let P(S) denote the power set of S = \ 1,2,3, ... ,10\. Define the relations R_1 and R_2 on P(S) as AR_1B if ( A intersection B^C ) union ( B intersection A^C ) = phi and AR_2B if union B^C = B union A^C,forall A,B in P(S). Then

DivineJEE · Accepted Answer

AR_1 B => A intersection B^ C union B intersection A^ C=phi(i) for reflexive A R_1 A\ => A intersection A^ C union A intersection A^ C=phi union phi=phi(ii) symmetric\ AR_1 B => BR_1 A => A intersection B^ C union B intersection A^ C=phi => AR_1 B => B intersection A^ C union A intersection B^ C=phi => BR_1 ATherefore It is symmetric(iii) Transitive AR_1 B => A intersection B^ C union B intersection A^ C=phi BR_1 C => B intersection C^ C union C intersection B^ C=phi => A intersection B^ C=phi \ B intersection A^ C=phi B intersection C^ C=phi \ C intersection B^ C=phi => A intersection C^ C=phi \ C intersection A^ C=phiTherefore R_1 is transitiveTherefore R_1 is equivalence relation.R_2: A R_2 B => A union B^C=B union A^C A union B^C=mu-(B-A)Similarly B union A^ C=mu-( A- B)If AR_2 B => A- B= B- A i) Reflexive A- A= A- A ii) Symmetric\ AR_2 B => A- B= B- A BR_2 A => B- A= A- B iii) Transitive\ AR_2 B => A- B= B- A and BR_2 C => B- C= C- B\ (1)+(2) => A- C= C- A => AR_2 C => CR_2 A\ Therefore R_2 is also equivalence

Explanation

The minimum number of elements that must be added to the relation R = {(a,b),(b,c),(b,d)} on the set { a,b,c,d} so that it is an equivalence relation is

Let R be the set of real numbers. end{enumerate} Statement-1: A = {(x,y) in R X R:y - x is an integer } is an equivalence . relation on R. Statement-2: B = {(x,y) in R X R:x = alpha y for some rational number } is . an equivalence relation on R.

Let A = { 1,3,4,6,9} and B = { 2,4,5,8,10}. Let R be a relation defined on A X B such that R = {( ( a_{1},b_{1} ), ( a_{2},b_{2} )):a_{1} ? b_{2} and b_{1} ? a_{2}}. Then the number of elements in the set R is

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The number of relations, on the set { 1,2,3} containing (1,2) and (2,3), which are reflexive and transitive but not symmetric, is .

Let A = { 0,3,4,6,7,8,9,10} and R be the relation defined on A such that R = {(x,y) in A X A:x - y is odd positive integer or x - y = 2}. The minimum number of elements that must be added to the relation R, so that it is a symmetric relation, is equal to .

If R is the smallest equivalence relation on the set { 1,2,3,4} such that {(1,2),(1,3)} subset R, then the number of elements in R is .

Let X = { 1,2,3,4,5}. The number of different ordered pairs ( Y,Z ) that can be formed such that Y subseteq X,Z subseteq X and Y cap Z. pty is

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