JEE Main 2023 — Mathematics Algebra

We have, $A={1,2,3,4,5,6,7}$A=1,2,3,4,5,6,7

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

Thus, $R\mathrm{is}\mathrm{reflexive}\Leftrightarrow (a,a)\in R\mathrm{for}\mathrm{all}a\in A$Risreflexive⇔(a,a)∈Rforalla∈A

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

Hence $R$R is not reflexive.

Symmetric: A relation $R$R is symmetric on a set $A$A iff

$$(a,b)\in R\Rightarrow (b,a)\in R\mathrm{for}\mathrm{all}a,b\in A$$(a,b)∈R⇒(b,a)∈Rforalla,b∈A

$$\Rightarrow x+y=7$$⇒x+y=7

Now on interchanging $y$y and $x$x we get, $y+x=7$y+x=7 which is always true for given set,

Hence $R$R is symmetric.

Transitive: A relation $R$R on $A$A is said to be transitive relation iff

$$(a,b)\in R\mathrm{and}(b,c)\in R$$(a,b)∈Rand(b,c)∈R

$$\Rightarrow (a,c)\in R\mathrm{for}\mathrm{all}a,b,c\in A$$⇒(a,c)∈Rforalla,b,c∈A

Now taking $(a,b)\equiv (3,4)$(a,b)≡(3,4) and $(b,c)\equiv (4,3)$(b,c)≡(4,3) so $(a,c)\equiv (3,3)$(a,c)≡(3,3) does not satisfy $x+y=7$x+y=7,

Hence, $R$R is not transitive and not equivalence.

Therefore, $R$R is only Symmetric.