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Maths Question 2 – JEE-MAIN 2025

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Let A={−3,−2,−1,0,1,2,3} and R be a relation on A defined by xRy if and only if 2x−y∈{0,1}. Let l be the number of elements in R. Let m and n be the minimum number of elements required to be added in R to make it reflexive and symmetric relations, respectively. Then l+m+n is equal to :

First, determine all ordered pairs (x,y) that satisfy the given condition 2x−y∈{0,1} where both x and y belong to the set A.

Step 1: Determine the elements of relation R and calculate l✦ Active

The set is A={−3,−2,−1,0,1,2,3}. The relation R is defined by xRy if 2x−y=0 or 2x−y=1. This means y=2x or y=2x−1. We find all pairs (x,y) such that x,y∈A:

For y=2x: (−1,−2),(0,0),(1,2) For y=2x−1: (−1,−3),(0,−1),(1,1),(2,3) R={(−1,−2),(0,0),(1,2),(−1,−3),(0,−1),(1,1),(2,3)} l=|R|=7
Step 2: Calculate m for reflexivity○ Expand

For R to be reflexive, all elements (a,a) for a∈A must be in R. The diagonal elements are D={(−3,−3),(−2,−2),(−1,−1),(0,0),(1,1),(2,2),(3,3)}. The elements from D already in R are (0,0) and (1,1). The elements needed to be added are D∖R:

D∖R={(−3,−3),(−2,−2),(−1,−1),(2,2),(3,3)} m=|D∖R|=5
💡 Teacher's Secret Hint

Remember that m is the *minimum* number of elements to *add*, so only count the missing diagonal pairs.

Step 3: Calculate n for symmetry and final sum○ Expand

For R to be symmetric, if (x,y)∈R, then (y,x) must also be in R. We check the pairs in R and identify missing symmetric counterparts:

(−1,−2)∈R⟹add (−2,−1) (1,2)∈R⟹add (2,1) (−1,−3)∈R⟹add (−3,−1) (0,−1)∈R⟹add (−1,0) (2,3)∈R⟹add (3,2) The pairs (0,0) and (1,1) are already symmetric. Number of elements to add for symmetry, n=5

Finally, we calculate l+m+n:

l+m+n=7+5+5=17
💡 Teacher's Secret Hint

Ensure you only count pairs (y,x) that are not already present in R when calculating n.

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