Question

Let $P(S)$ denote the power set of $S=\\{1,2,3, \ldots , 10\\}$. Define the relations $R_1$ and $R_2$ on $P(S)$ as $A_1 B$ if $\left(A \cap B^c\right) \cup\left(B \cap A^c\right)=\emptyset$ and $A_2 B$ if $A \cup B^c=B \cup A^c, \forall A, B \in P(S)$. Then :

• A. both $R_1$ and $R_2$ are not equivalence relations
• B. only $R_2$ is an equivalence relation
• C. only $R_1$ is an equivalence relation
• D. both $R_1$ and $R_2$ are equivalence relations

Answer 1

Answer: (A)

both $R_1$ and $R_2$ are not equivalence relations

Answer 2

S={1,2,3,……10}
P(S)= power set of S
AR,B⇒(A∩B)∪(A∩B)=ϕ
R1 is reflexive, symmetric
For transitive
(A∩B)∪(A∩B)=ϕ;{a}=ϕ={b}A=B
(B∩C)∪(B∩C)=ϕ∴B=C
∴A=C equivalence.

R2​≡A∪B=A∪B
R2​→ Reflexive, symmetric for transitive

A∪B=A∪B⇒{a,c,d}={b,c,d}
{a}={b}
∴A=B
B∪C=B∪C⇒B=C
∴A=C
∴A∪C=A∪C
∴ Equivalence