Let P ( S ) denote the power set of S ={1,2,3 | vedclass

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Question

$S=\{1,2,3, \ldots \ldots .10\}$

$P ( S )=$ power set of $S$

$AR , B \Rightarrow( A \cap \overrightarrow{ B }) \cup(\overrightarrow{ A } \cap B

Vedclass · Accepted Answer

both $R_1$ and $R_2$ are equivalence relations

Vedclass · Answer

$S=\{1,2,3, \ldots \ldots .10\}$

$P ( S )=$ power set of $S$

$AR , B \Rightarrow( A \cap \overrightarrow{ B }) \cup(\overrightarrow{ A } \cap B )=\phi$

$R 1$ is reflexive, symmetric

For transitive

$( A \cap \overrightarrow{ B }) \cup(\overrightarrow{ A } \cap B )=\phi ;\{ a \}=\phi=\{ b \} A = B$

$( B \cap \overrightarrow{ C }) \cup(\overrightarrow{ B } \cap C )=\phi 	herefore B = C 	herefore A = C 	ext { equivalence. }$

$R _2 \equiv A \cup \overrightarrow{ B }=\overrightarrow{ A } \cup B$

$R _2 ightarrow 	ext { Reflexive, symmetric }$

for transitive

$\begin{array}{l} A \cup \overrightarrow{ B }=\overrightarrow{ A } \cup B \Rightarrow\{ a , c , d \}=\{ b , c , d \} \ \{ a \}=\{ b \} 	herefore A = B \ B \cup \overrightarrow{ C }=\overrightarrow{ B } \cup C \Rightarrow B = C \ 	herefore A = C \quad 	herefore A \cup \overrightarrow{ C }=\overrightarrow{ A } \cup C 	herefore 	ext { Equivalence }\end{array}$

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 $A R_1 B$ if $\left( A \cap B ^{ c }\right) \cup\left( B \cap A ^{ c }\right)=\varnothing$ and $AR _2 B$ if $A \cup B ^{ c }=$ $B \cup A ^{ c }, \forall A , B \in P ( S )$. Then :

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