Question

Let $P(S)$ denote the power set of $S=\{1,2,3,\dots .,10\}$. Define the relations $R_1$ and $R_2$ on $P(S)$ as $A_{1}B$ if $\left(A\cap B^c\right)\cup \left(B\cap A^c\right)=\varnothing$ 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)

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

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

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