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Count a relation's elements and the minimum to add to make it reflexive or symmetric

This type of question, asked 9 times: 8 Apr 2023, 13 Apr 2023, 31 Jan 2024, 6 Apr 2024, 3 Apr 2025, 4 Apr 2025, 21 Jan 2026 S2, 22 Jan 2026 S1, 23 Jan 2026 S1. All 11 questions below, oldest first.

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  1. Question 1

    8 Apr 2023

    86
    Let A={0,3,4,6,7,8,9,10}\mathbf{A = \{ 0,3,4,6,7,8,9,10\}} and RR be the relation defined on A such that R={(x,y)∈A×A:x−y \mathbf{R = \{(x,y) \in A \times A:x - y\ } is odd positive integer or x−y=2}x - y = 2\}. The minimum number of elements that must be added to the relation RR, so that it is a symmetric relation, is equal to

  2. Question 2

    13 Apr 2023

    172
    Let A={−4,−3,−2,0,1,3,4}\mathbf{A = \{ - 4, - 3, - 2,0,1,3,4\}} and R={(a,b)∈A\mathbf{R = \{(a,b) \in A} ×A:b=∣a∣\times A:b = |a| or  b2=a+1}\left. \ b^{2}= a + 1 \right\} be a relation on AA. Then the minimum number of elements, that must be added to the relation RR so that it becomes reflexive and symmetric, is

  3. Question 3

    31 Jan 2024

    180
    Let A={1,2,3,…….100}\mathbf{A = \{ 1,2,3,\ldots\ldots.100\}}. Let RR be a relation on A defined by (x,y)∈R(x,y) \in R if and only if 2x=3y2x= 3y. Let R1R_{1} be a symmetric relation on AA such that R⊂R1R \subset R_{1} and the number of elements in R1R_{1} is nn. Then, the minimum value of nn is

  4. Question 4

    6 Apr 2024

    72
    Let the relations R1R_{1} and R2R_{2} on the set X={1,2,3,…,20}\mathbf{X = \{ 1,2,3,\ldots,20\}} be given by R1={(x,y):2x−3y=2}R_{1}\mathbf{= \{(x,y):2x - 3y = 2\}} and R2={(x,y):−5x+4y=0}R_{2}\mathbf{= \{(x,y): - 5x + 4y = 0\}}. If MM and NN be the minimum number of elements required to be added in R1R_{1} and R2R_{2}, respectively, in order to make the relations symmetric, then M+NM + N equals

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  5. Question 5

    6 Apr 2024

    152
    Let A={1,2,3,4,5}\mathbf{A = \{ 1,2,3,4,5\}}. Let RR be a relation on AA defined by xRyxRy if and only if 4x≤5y4x \leq 5y. Let mm be the number of elements in RR and nn be the minimum number of elements from A×AA \times A that are required to be added to RR to make it a symmetric relation. Then m+nm + n is equal to:

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  6. Question 6

    3 Apr 2025

    131
    Let A={−2,−1,0,1,2,3}A=\left\{ - 2, - 1,0,1,2,3 \right\}. let RR be a relation on A defined by xRyxRy if and only if y=max{x,1}y\mathbf{= max\{}x,1\}. Let ll be the number of elements in R . Let m and n be the minimum number of elements required to be added in RR to make it reflexive and symmetric relations, respectively. Then l+m+nl+ m + n is equal to.

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  7. Question 7

    3 Apr 2025

    55
    Let A={−3,−2,−1,0,1,2,3}A=\{-3,-2,-1,0,1,2,3\}. Let RR be a relation on AA defined by xRyxRy if and only if 0≤x2+2y≤40\leq x^{2}+2y\leq 4. Let ll be the number of elements in RR and mm be the minimum number of elements required to be added in RR to make it a reflexive relation. Then l+ml+m is equal to.

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  8. Question 8

    4 Apr 2025

    129
    Let A={−3,−2,−1,0,1,2,3}\mathbf{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}\mathbf{2x - y \in \{ 0,1\}}. Let ll 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+mnl+ mn is equal to:-

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  9. Question 9

    21 Jan 2026 S2

    67
    Let A={2,3,5,7,9}\mathbf{A = \{ 2,3,5,7,9\}}. Let R be the relation on A defined by x Ry if and only if 2x≤3y2x \leq 3y. Let l\mathcal{l} be the number of elements in R , and m be the minimum number of elements required to be added in R to make it a symmetric relation. Then l+m\mathcal{l +}m is equal to:

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  10. Question 10

    22 Jan 2026 S1

    52
    Let the relation R on the set M={1,2,3,…….16}\mathbf{M = \{ 1,2,3,\ldots\ldots.16\}} be given by R={(x,y):4y=5x−3,x,y∈M}\mathbf{R = \{(x,y):4y = 5x - 3,x,y \in M\}}. Then the minimum number of elements required to be added in R , in order to make the relation symmetric, is equal to

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  11. Question 11

    23 Jan 2026 S1

    56
    Let A={−2,−1,0,1,2,3,4}\mathbf{A = \{ - 2, - 1,0,1,2,3,4\}}. Let R be a relation on A defined by xRy if and only if 2x+y≤22x + y \leq 2. Let ll 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+nl + m + n is equal to :

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