જો R અને S એ ગણ A પરના સામ્ય સંબંધ હોય તો | vedclass

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Question

(b) Given, $R$ and $S$ are relations on set $A$.

 $	herefore R \subseteq A 	imes A$and $S \subseteq A 	imes A$ ==> $R \cap C \subseteq A

Vedclass · Accepted Answer

$R \cap S $ એ  $A$ પરનો સામ્ય સંબંધ છે.

Vedclass · Answer

(b) Given, $R$ and $S$ are relations on set $A$.

 $	herefore R \subseteq A 	imes A$and $S \subseteq A 	imes A$ ==> $R \cap C \subseteq A 	imes A$

==> $R \cap S$is also a relation on $A$.

Reflexivity : Let a be an arbitrary element of $A$. Then, $a \in A$==> $(a,a) \in R$and $(a,a) \in S$,

[$	herefore$ $R$ and $S$ are reflexive]

==> $(a,\,a) \in R \cap S$

Thus, $(a,a) \in R \cap S$for all $a \in A$.

So, $R \cap S$is a reflexive relation on $ A$.

Symmetry : Let $a,b \in A$such that $(a,b) \in R \cap S$.

Then, $(a,b) \in R \cap S$ ==> $(a,b) \in R$and $(a,b) \in S$

==> $(b,a) \in R$ and $(b,a) \in S$,

[ $	herefore $ $R$ and $S$ are symmetric]

==> $(b,a) \in R \cap S$

Thus, $(a,b) \in R \cap S$

==> $(b,a) \in R \cap S$ for all $(a,b) \in R \cap S$.

So, $R \cap S$ is symmetric on $A$.

Transitivity : Let $a,b,c \in A$ such that $(a,b) \in R \cap S$ and $(b,c) \in R \cap S$. 

Then, $(a,b) \in R \cap S$ and $(b,c) \in R \cap S$

==> $\{ ((a,b) \in R\,\,{m{and}}\,(a,b) \in S\,)\} $

and $\{ ((b,c) \in R\,{m{and}}\,{m{(}}b,c{m{)}} \in S\} $

==> $\{ (a,b) \in R,(b,c) \in R\} $ and $\{ (a,b) \in S,(b,c) \in S\} $

==> $(a,c) \in R$ and $(a,c) \in S$

$\left[ \begin{gathered}
  \because R\,{	ext{and}}\,S\,{	ext{are}}\,{	ext{transitive}}\,{	ext{So}} \hfill \
  {	ext{(}}a,b{	ext{)}} \in R\,{	ext{and (}}b,c{	ext{)}} \in R \Rightarrow {	ext{(}}a{	ext{,}}c{	ext{)}} \in R \hfill \
  {	ext{(}}a,b{	ext{)}} \in S{	ext{ and (}}b{	ext{,}}c{	ext{)}} \in S \Rightarrow {	ext{(}}a{	ext{,}}c{	ext{)}} \in S \hfill \ 
\end{gathered}  ight.$

==> $(a,c) \in R \cap S$

Thus,$(a,b) \in R \cap S$ and$(b,c) \in R \cap S \Rightarrow (a,c) \in R \cap S$.

So, $R \cap S$ is transitive on $A$.

Hence, $R$ is an equivalence relation on $A$.

જો $R$ અને $S$ એ ગણ $A$ પરના સામ્ય સંબંધ હોય તો

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