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

The Correct Option is A

$S=\{1,2,3,\dots \dots 10\}$
$P(S)=\text{power set of}S$
$AR,B\Rightarrow (A\cap \overrightarrow{B})\cup (\overrightarrow{A}\cap B)=\varphi$
$R1\text{is reflexive, symmetric}$
$\text{For transitive}$
$(A\cap \overrightarrow{B})\cup (\overrightarrow{A}\cap B)=\varphi ;\{a\}=\varphi =\{b\}A=B$
$(B\cap \overrightarrow{C})\cup (\overrightarrow{B}\cap C)=\varphi \therefore B=C$
$\therefore A=C\text{equivalence.}$

$R_2\equiv A\cup \overrightarrow{B}=\overrightarrow{A}\cup B$
$R_2\to$ Reflexive, symmetric for transitive

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