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JEE Main 20238 AprilEvening Shift
Exam
JEE Main
Year
2023
Date
8 April
Shift
Evening Shift
Subject
Mathematics
Chapter
Sets and Relations

JEE Main 2023 · 8 April · Evening Shift · Mathematics · Sets and Relations

Let A = 1, 2, 3, 4, 5, 6, 7 . Then the relation R = ( x, y ) ∈ A ×...

Let A=1,2,3,4,5,6,7A = { 1, 2, 3, 4, 5, 6, 7 }. Then the relation R=(x,y)∈A×A:x+y=7R = { ( x, y ) \in A \times A: x + y = 7 } is

Options

  1. Aan equivalence relation
  2. Bsymmetric but neither reflexive nor transitive
  3. Ctransitive but neither symmetric nor reflexive
  4. Dreflexive but neither symmetric nor transitive

Correct answer

B. symmetric but neither reflexive nor transitive

Step-by-step solution

We have, A = {1,2,3,4,5,6,7}\left\{1, 2, 3, 4, 5, 6, 7\right\}

Reflexive: A relation RR on a set AissaidtobereflexiveifeveryelementofAis said to be reflexive if every element of A is related to itself.

Thus, Risreflexive⇔(a,a)∈Rforalla∈ARisreflexive ⇔ ( a, a ) \in Rforalla \in A

∵(1,1),(2,2),(3,3),......(7,7)∵ ( 1, 1 ) , \left(2, 2\right), ( 3, 3 ) , . . . . . . ( 7, 7 ) does not satisfy x+y=7x + y = 7

Hence RR is not reflexive.

Symmetric: A relation RR is symmetric on a set Aiff</p><p>iff</p><p>( a, b ) \in R ⇒ ( b, a ) \in Rforalla, b \in A</p><p></p><p>⇒ x + y = 7</p><p>Nowoninterchanging</p><p>Now on interchangingy and xweget,we get,y + x = 7whichisalwaystrueforgivenset,</p><p>Hencewhich is always true for given set,</p><p>HenceRissymmetric.</p><p><strong>Transitive</strong>:Arelationis symmetric.</p><p><strong>Transitive</strong>: A relationRonAon A is said to be transitive relation iff

\left(a, b\right) \in Rand ( b, c ) \in R</p><p></p><p>⇒ \left(a, c\right) \in Rforalla, b, c \in A</p><p>Nowtaking(a,b)≡(3,4)</p><p>Now taking \left(a, b\right) ≡ \left(3, 4\right) and \left(b, c\right) ≡\left(4, 3\right) so \left(a, c\right) ≡ \left(3, 3\right) does not satisfyx+y=7x + y = 7,

Hence, RR is not transitive and not equivalence.

Therefore, R is only Symmetric.

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