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)

$ S =\{1,2,3, \ldots \ldots 10\} $
$P ( S )=\text { power set of } S $
$ AR , B \Rightarrow( A \cap \vec{ B }) \cup(\vec{ A } \cap B )=\phi $
$ R 1 \text { is reflexive, symmetric } $
$ \text { For transitive } $
$ ( A \cap \vec{ B }) \cup(\vec{ A } \cap B )=\phi ;\{ a \}=\phi=\{ b \} A = B $
$ ( B \cap \vec{ C }) \cup(\vec{ B } \cap C )=\phi \therefore B = C $
$ \therefore A = C \text { equivalence. }$

$ R _2 \equiv A \cup \vec{ B }=\vec{ A } \cup B$
$ R _2 \rightarrow $ Reflexive, symmetric for transitive

$ A \cup \vec{ B }=\vec{ A } \cup B \Rightarrow\{ a , c , d \}=\{ b , c , d \} $
$ \{ a \}=\{ b \}$
$ \therefore A = B $
$ B \cup \vec{ C }=\vec{ B } \cup C \Rightarrow B = C$
$\therefore A = C $
$\therefore A \cup \vec{ C }=\vec{ A } \cup C $
$\therefore \text { Equivalence }$