Question

lf A = {x ∈ z⁺ : x < 10 and x is a multiple of 3 or 4}, where z⁺ is the set of positive integers, then the total number of symmetric relations on A is ______.

Options

• 2⁵

• 2¹⁵

• 2¹⁰

• 2²⁰

Answer 1

lf A = {x ∈ z⁺ : x < 10 and x is a multiple of 3 or 4}, where z⁺ is the set of positive integers, then the total number of symmetric relations on A is `underlinebb(2^15)`.

Explanation:

A relation on a set A is said to be symmetric if (a, b) ∈ A `\implies` (b, a) ∈ A, ∀ a, b ∈ A

Here A = {3, 4, 6, 8, 9}

Number of order pairs of A × A = 5 × 5 = 25

Divide 25 order pairs of A × A into 3 parts as follows:

Part – A: (3, 3), (4, 4), (6, 6), (8, 8), (9, 9)

Part – B: (3, 4), (3, 6), (3, 8), (3, 9), (4, 6), (4, 8), (4, 9), (6, 8), (6, 9), (8, 9)

Part – C: (4, 3), (6, 3), (8, 3), (9, 3), (6, 4), (8, 4), (9, 4), (8, 6), (9, 6), (9, 8)

In part – A, both components of each ordered pair are the same.

In part – B, both components are different but not two such order pairs are present in which the first component of one order pair is the second component of another order pair and vice-versa.

In part – C, only the reverse of the order pairs of part – B are present i.e., if (a, b) is present in part – B, then (b, a) will be present in part – C.

For example (3, 4) is present in part – B, and (4, 3) is present in part – C.

Number of order pairs in A, B and C are 5, 10 and 10 respectively.

In any symmetric relation on set A, if any ordered pair of part – B is present then its reverse order pair of part – C will be also present.

Hence the number of symmetric relations on set A is equal to the number of all relations on set D, which contains all the order pairs of part – A and part – B.

Now n(D) = n(A) + n(B) = 5 + 10 = 15

Hence the number of all relations on set D = (2)¹⁵

`\implies` Number of symmetric relations on set D = (2)¹⁵