JEE Main 2022 — Sets And Relations Question with Solution

$$\sum _{i=1}^{10}\sum _{j=1}^{10}\{i,j\}$$

= {1, 1} {1, 2} {1, 3} ..... {1, 10}

{2, 1} {2, 2} {2, 3} ..... {2, 10}

{3, 1} {3, 2} {3, 3} ..... {3, 10}

$$⋮$$

{10, 1} {10, 2} {10, 3} ..... {10, 10}

Now, $$A=\sum _{i=1}^{10}\sum _{j=1}^{10}min\{i,j\}$$

= minimum between i and j in all sets and summation of all those values.

and $$B=\sum _{i=1}^{10}\sum _{j=1}^{10}\max \{i,j\}$$

= maximum between i and j in all sets and summation of all those values.

For 1 :

1 is minimum in sets =

{1, 1}, {1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {1, 8}, {1, 9}, {1, 10}, {2, 1}, {3, 1}, {4, 1}, {5, 1}, {6, 1}, {7, 1}, {8, 1}, {9, 1}, {10, 1}

$$\therefore$$ 1 is minimum in 19 sets

1 is maximum in {1, 1} sets.

$$\therefore$$ 1 is maximum and minimum in total 20 sets.

$$\therefore$$ Sum of 1 in all those sets = 1 $$\times$$ 20 = 20

For 2 :

2 is minimum in sets =

{2, 2}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {2, 7}, {2, 8}, {2, 9}, {2, 10}, {3, 2}, {4, 2}, {5, 2}, {6, 2}, {7, 2}, {8, 2}, {9, 2}, {10, 2}

$$\therefore$$ 2 is minimum in 17 sets

2 is maximum in sets = {1, 2}, {2, 1}, {2, 2}

$$\therefore$$ 2 is maximum and minimum in 20 sets.

$$\therefore$$ Sum of 2 in all those sets = 2 $$\times$$ 20 = 40

Similarly 3 is maximum and minimum in 20 sets.

$$\therefore$$ Sum of 3 in all those sets = 20 $$\times$$ 3 = 60

$$⋮$$

Similarly, 10 is maximum and minimum in 20 sets.

$$\therefore$$ Sum of 10 in all those sets = 20 $$\times$$ 10 = 200

$$\therefore$$ A + B = 20 + 20 $$\times$$ 2 + 20 $$\times$$ 3 + ....... + 20 $$\times$$ 10

= 20(1 + 2 + 3 + ...... + 10)

= 20 $$\times$$ $$\frac{10\times 11}{2}$$

= 1100