Let P (S) denote the power set of S = \ 1, 2, 3, , 10\ . Define the relations R _ 1 and R _ 2 on P(S) as A R _ 1 B if (A B ^ c ) (B A ^ c )= φ and A R _ 2 B if A B ^ c = B A ^ c , A, B ∈ P(S) . Then :

Correct answer: A. both R _ 1 and R _ 2 are equivalence relations Given, S = \ 1, 2, 3, 10\ P(S) = power set of S A R _ 1 B ⇒ (A B ^ c ) ( A ^ c B) = ϕ Now for reflexive property, replacing B with A we get, (A A ^ c ) ( A ^ c A) which ϕ always, Now checking symmetric we will interchange A and B , So, (B A ^ c ) ( B ^ c A) which is same as (A B ^ c ) ( A ^ c B) , hence the relation is symmetric, So, R _ 1 is reflexive, symmetric Now checking for transitive (A B ^ c ) ( A ^ c B) = ϕ; Now from diagram the elements in (A B ^ c ) ( A ^ c B) will be, \ a\ \ b\ which is given as empty set ϕ Hence, we can say that, \ a\ = ϕ = \ b\ ⇒ A = B Now taking, ( B C ^ c ) ( B ^ c C ) = ϕ ∴ B = C ∴ A = C equivalence. Now solving, R _ 2 ≡ A B ^ c = A ^ c B Now for reflexive replacing B → A we get, A A ^ c = A ^ c A which is true, And for symmetric interchanging A ⇔ B we get, B A ^ c = B ^ c A which is again true, Hence, R _ 2 → Reflexive, symmetric Now for transitive, From diagram the elements in A B ^ c = A ^ c B ⇒ \ a, c, d\ = \ b, c, d\ On comparing both side, we get \ a\ = \ b\ ∴ A = B And, B C ^ c = B ^ c C ⇒ B = C ∴ A = C ∴ A C ^ c = A ^ c C ∴ Equivalence Hence, both given relation are equivalence.

JEE Main 20231 FebruaryEvening Shift

Exam: JEE Main
Year: 2023
Date: 1 February
Shift: Evening Shift
Subject: Mathematics
Chapter: Sets and Relations

JEE Main 2023 · 1 February · Evening Shift · Mathematics · Sets and Relations

Let P (S) denote the power set of S = \ 1, 2, 3, , 10\ . Define the...

Let

P

\left(S\right)

denote the power set of S = \left\{1, 2, 3, \ldots , 10\right\}

. Define the relations

{R}_{1}

and

{R}_{2}

P\left(S\right)

A{R}_{1}B

\left(A \cap {B}^{c}\right)

\cup \left(B \cap {A}^{c}\right)= \phi

and

A{R}_{2}B

if

A \cup {B}^{c} = B \cup {A}^{c}

, \forall A, B \in P\left(S\right)

. Then : $Let P (S) denote the power set of S = \ 1, 2, 3, , 10\ . Define the relations R _ 1 and R _ 2 on P(S) as A R _ 1 B if$ $Let P (S) denote the power set of S = \ 1, 2, 3, , 10\ . Define the relations R _ 1 and R _ 2 on P(S) as A R _ 1 B if$

Options

Aboth ${R}_{1}$ and ${R}_{2}$ are equivalence relations
Bonly ${R}_{1}$ is an equivalence relation
Conly ${R}_{2}$ is an equivalence relation
Dboth ${R}_{1}$ and ${R}_{2}$ are not equivalence relations

Correct answer

A. both ${R}_{1}$ and ${R}_{2}$ are equivalence relations

Step-by-step solution

Given,

$S = \left\{1, 2, 3, \ldots \ldots 10\right\}$

$P\left(S\right) =$ power set of $S$

$A{R}_{1}B ⇒ \left(A \cap {B}^{c}\right) \cup \left({A}^{c} \cap B\right) = ϕ$

Now for reflexive property, replacing $B$ with $A$ we get,

\left(A \cap {A}^{c}\right) \cup \left({A}^{c} \cap A\right) $which$ ϕ $always,Now checking symmetric we will interchange$ A \text{ and } B $,So,$ \left(B \cap {A}^{c}\right) \cup $\left({B}^{c} \cap A\right)$ which is same as \left(A \cap {B}^{c}\right) \cup \left({A}^{c} \cap B\right) $, hence the relation is symmetric,So,$ {R}_{1} $is reflexive, symmetricNow checking for transitive \left(A \cap {B}^{c}\right) \cup \left({A}^{c} \cap B\right) = ϕ;$

Now from diagram the elements in \left(A \cap {B}^{c}\right) \cup \left({A}^{c} \cap B\right) $will be,\left\{a\right\} \cup \left\{b\right\}$ which is given as empty set $ϕ$

Hence, we can say that, \left\{a\right\} = ϕ = \left\{b\right\} ⇒ A = B $Now taking,$ ( B \cap {C}^{c} ) \cup ( {B}^{c} \cap C ) = ϕ ∴ B = C $$ ∴ A = C $equivalence.<img src="https://cdn-question-pool.Quantrex.app/pyq/jee_main/7dda9698-159e-473c-a185-7b6eb321b64c-9896795.png" style="width: 250px; height: 219px;" /> Now solving,$ {R}_{2} ≡ A \cup {B}^{c} = {A}^{c} \cup B $Now for reflexive replacing$ B \to A $we get,$ A \cup {A}^{c} = {A}^{c} \cup A $which is true,And for symmetric interchanging$ A ⇔ B $we get,$ B \cup {A}^{c} = {B}^{c} \cup A $which is again true,Hence,$ {R}_{2} \to $Reflexive, symmetric<img src="https://cdn-question-pool.Quantrex.app/pyq/jee_main/f85793dd-c45c-48ad-94cd-a80954a303c0-9896795_2.png" style="width: 250px; height: 219px;" />Now for transitive,From diagram the elements in$ A \cup {B}^{c} = {A}^{c} \cup B ⇒ \left\{a, c, d\right\} = \left\{b, c, d\right\} $On comparing both side, we get \left\{a\right\} = \left\{b\right\} ∴ A = B$

And, $B \cup {C}^{c} = {B}^{c} \cup C ⇒ B = C$

$∴ A = C ∴ A \cup {C}^{c} = {A}^{c} \cup C$

$∴$ Equivalence

Hence, both given relation are equivalence.

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