Maths pyq jee mains 2023

Question

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

(1) Symmetric but neither reflexive nor transitive

(2) Transitive but neither symmetric nor reflexive

(3) An equivalence relation

(4) Reflexive but neither symmetric nor transitive

Answer 1

The relation R={(x,y)∈A×A:x+y=7}R = \{(x, y) \in A \times A : x + y = 7\}R={(x,y)∈A×A:x+y=7} has the following properties:

1. Not Reflexive: Reflexivity requires (x,x)∈R(x, x) \in R(x,x)∈R, but x+x=7x + x = 7x+x=7 is not true for any x∈Ax \in Ax∈A.
2. Symmetric: If (x,y)∈R(x, y) \in R(x,y)∈R, then x+y=7x + y = 7x+y=7 implies y+x=7y + x = 7y+x=7, so (y,x)∈R(y, x) \in R(y,x)∈R.
3. Not Transitive: If (x,y)∈R(x, y) \in R(x,y)∈R and (y,z)∈R(y, z) \in R(y,z)∈R, it does not necessarily follow that (x,z)∈R(x, z) \in R(x,z)∈R, since x+z≠7x + z \neq 7x+z=7 in general.

Thus, the relation is symmetric but neither reflexive nor transitive.
Answer: (1) Symmetric but neither reflexive nor transitive.