The relation R and R′ are symmetric in the set A, then show that R∪R′ and R∩R′ are symmetric.

We will prove, union of sets is symmetric Let S=R∪R′Let (a,b)∈S⇒(a,b)∈R∪R′⇒(a,b)∈R or (a,b)∈R′⇒(b,a)∈Ror (b,a)∈R′ (Since, R and R' are symmetric)⇒(b,a)∈R∪R′⇒(b,a)∈SHence, S=R∪R′ is symmetric.Now, we will prove, intersection of sets is symmetric Let P=R∩R′Let (a,b)∈P⇒(a,b)∈R∩R′⇒(a,b)∈R and (a,b)∈R′⇒(b,a)∈R and (b,a)∈R′ (Since, R and R' are symmetric)⇒(b,a)∈R∩R′⇒(b,a)∈PHence, P=R∩R′ is symmetric.

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The relation R and R' are symmetric in the set A, then show th

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The relation $R$ and $R^{'}$ are symmetric in the set $A$ , then show that $R \cup R^{'}$ and $R \cap R^{'}$ are symmetric.

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The relation

R

and

R^{'}

are symmetric in the set

A

, then show that

R \cup R^{'}

and

R \cap R^{'}

are symmetric.

9 mins ago

Discuss this question LIVE

9 mins ago

Text solutionVerified

We will prove, union of sets is symmetric
Let

S = R \cup R^{'}

Let

(a, b) \in S

\Rightarrow (a, b) \in R \cup R^{'}

\Rightarrow (a, b) \in R

(a, b) \in R^{'}

\Rightarrow (b, a) \in R

(b, a) \in R^{'}

(Since, R and R' are symmetric)

\Rightarrow (b, a) \in R \cup R^{'}

\Rightarrow (b, a) \in S

Hence,

S = R \cup R^{'}

is symmetric.

Now, we will prove, intersection of sets is symmetric

Let

P = R \cap R^{'}

Let

(a, b) \in P

\Rightarrow (a, b) \in R \cap R^{'}

\Rightarrow (a, b) \in R

and

(a, b) \in R^{'}

\Rightarrow (b, a) \in R

and

(b, a) \in R^{'}

(Since, R and R' are symmetric)

\Rightarrow (b, a) \in R \cap R^{'}

\Rightarrow (b, a) \in P

Hence,

P = R \cap R^{'}

is symmetric.

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Question Text	The relation $R$ and $R^{'}$ are symmetric in the set $A$ , then show that $R \cup R^{'}$ and $R \cap R^{'}$ are symmetric.
Answer Type	Text solution:1
Upvotes	150

The relation R and R′ are symmetric in the set A, then show that R∪R′ and R∩R′ are symmetric.

Text solutionVerified

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The relation $R$ and $R^{'}$ are symmetric in the set $A$ , then show that $R \cup R^{'}$ and $R \cap R^{'}$ are symmetric.