Let A = {2, 3, 4, 5, 6}. Let R be a relation on the set A x A given by (x, y) R (z, w) if and only if x divides z and y w. Then the number of elements in R is ________. [2026]

Question

Let A = {2, 3, 4, 5, 6}. Let R be a relation on the set A x A given by (x, y) R (z, w) if and only if x divides z and y $\leq$ w. Then the number of elements in R is ________. [2026]

Smart Achievers · Accepted Answer

(120)

We have, A = {2, 3, 4, 5, 6}

We have, y $\leq$ w

$∙$ If y = 2, then w $\in$ {2, 3, 4, 5, 6}, Number of pairs = 5

$∙$ If y = 3, then w $\in$ {3, 4, 5, 6}, Number of pairs = 4

$∙$ If y = 5, then w $\in$ {4, 5, 6}, Number of pairs = 3

$∙$ If y = 5, then w $\in$ {5, 6}, Number of pairs = 2

$∙$ If y = 6, then w $\in$ {6}, Number of pairs = 1

Total pairs 5 + 4 + 3 + 2 + 1 = 15

Now, x divides z

$∴$ If x = 2, then z $\in$ {2, 4, 6}, Number of pairs = 3

If x = 3, then z $\in$ {3, 6}, Number of pairs = 2

If x = 4, then z $\in$ {4}, Number of pairs = 1

If x = 5, then z $\in$ {5}, Number of pairs = 1

If x = 6, then z $\in$ {6}, Number of pairs = 1

$∴$ Total pairs = 3 + 2 + 1 + 1 + 1 = 8

Total elements = 15 x 8 = 120.

Solution

Q.
Let A = {2, 3, 4, 5, 6}. Let R be a relation on the set A x A given by (x, y) R (z, w) if and only if x divides z and y $\leq$ w. Then the number of elements in R is ________. [2026]

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Solution

Q. Let A = {2, 3, 4, 5, 6}. Let R be a relation on the set A x A given by (x, y) R (z, w) if and only if x divides z and y ≤ w. Then the number of elements in R is ________. [2026]

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Q.
Let A = {2, 3, 4, 5, 6}. Let R be a relation on the set A x A given by (x, y) R (z, w) if and only if x divides z and y $\leq$ w. Then the number of elements in R is ________. [2026]