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Maths Question 1 – JEE-MAIN 2025

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Let A={−2,−1,0,1,2,3}. Let R be a relation on A defined by xRy if and only if y=max(x,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, explicitly list all ordered pairs (x,y) that satisfy the given relation y=max(x,1) for x∈A.

Step 1: Determine the initial relation R and its size l.✦ Active

The set is A={−2,−1,0,1,2,3}. The relation xRy is defined by y=max(x,1). We find the pairs (x,y) for each x∈A:

x=−2⟹y=max(−2,1)=1⟹(−2,1)∈R x=−1⟹y=max(−1,1)=1⟹(−1,1)∈R x=0⟹y=max(0,1)=1⟹(0,1)∈R x=1⟹y=max(1,1)=1⟹(1,1)∈R x=2⟹y=max(2,1)=2⟹(2,2)∈R x=3⟹y=max(3,1)=3⟹(3,3)∈R

Thus, R={(−2,1),(−1,1),(0,1),(1,1),(2,2),(3,3)}. The number of elements in R is l=|R|=6.

Step 2: Calculate m for reflexivity.○ Expand

For R to be reflexive, all pairs (a,a) for a∈A must be in R. The required diagonal elements are D={(−2,−2),(−1,−1),(0,0),(1,1),(2,2),(3,3)}. The elements from D already in R are {(1,1),(2,2),(3,3)}. The elements missing from R to make it reflexive are {(−2,−2),(−1,−1),(0,0)}. The minimum number of elements to add for reflexivity is m=3.

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

For R to be symmetric, if (a,b)∈R, then (b,a) must also be in R. Checking elements in R:

(−2,1)∈R, but (1,−2)∉R. Add (1,−2). (−1,1)∈R, but (1,−1)∉R. Add (1,−1). (0,1)∈R, but (1,0)∉R. Add (1,0). (1,1),(2,2),(3,3) are symmetric pairs of themselves.

The minimum number of elements to add for symmetry is n=3. Finally, l+m+n=6+3+3=12.

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