if A and B be two sets such that n(A)= 15 , n(B)= 25 then number of possible values of symmetric difference of A and B is

To find the number of possible values of the symmetric difference between two sets A and B, we first need to understand what symmetric difference means. The symmetric difference of two sets, denoted as A Δ B, consists of elements that are in either A or B, but not in both. Mathematically, it can be expressed as:

Definition of Symmetric Difference

A Δ B = (A - B) ∪ (B - A)

This means we take all elements that are unique to A and all elements that are unique to B.

Understanding Set Sizes

Given the sizes of the sets, we have:

• n(A) = 15 (meaning A has 15 elements)
• n(B) = 25 (meaning B has 25 elements)

Calculating the Symmetric Difference

The size of the symmetric difference can be calculated using the formula:

n(A Δ B) = n(A) + n(B) - 2n(A ∩ B)

Here, n(A ∩ B) represents the number of elements that are common to both sets A and B. The value of n(A ∩ B) can range from 0 (if A and B have no elements in common) to the minimum of n(A) and n(B), which is 15 in this case (if all elements of A are in B).

Range of Symmetric Difference

Now, let's explore the possible values of n(A Δ B) based on the range of n(A ∩ B):

• If n(A ∩ B) = 0: n(A Δ B) = 15 + 25 - 2(0) = 40
• If n(A ∩ B) = 1: n(A Δ B) = 15 + 25 - 2(1) = 38
• If n(A ∩ B) = 2: n(A Δ B) = 15 + 25 - 2(2) = 36
• If n(A ∩ B) = 3: n(A Δ B) = 15 + 25 - 2(3) = 34
• If n(A ∩ B) = 4: n(A Δ B) = 15 + 25 - 2(4) = 32
• If n(A ∩ B) = 5: n(A Δ B) = 15 + 25 - 2(5) = 30
• If n(A ∩ B) = 6: n(A Δ B) = 15 + 25 - 2(6) = 28
• If n(A ∩ B) = 7: n(A Δ B) = 15 + 25 - 2(7) = 26
• If n(A ∩ B) = 8: n(A Δ B) = 15 + 25 - 2(8) = 24
• If n(A ∩ B) = 9: n(A Δ B) = 15 + 25 - 2(9) = 22
• If n(A ∩ B) = 10: n(A Δ B) = 15 + 25 - 2(10) = 20
• If n(A ∩ B) = 11: n(A Δ B) = 15 + 25 - 2(11) = 18
• If n(A ∩ B) = 12: n(A Δ B) = 15 + 25 - 2(12) = 16
• If n(A ∩ B) = 13: n(A Δ B) = 15 + 25 - 2(13) = 14
• If n(A ∩ B) = 14: n(A Δ B) = 15 + 25 - 2(14) = 12
• If n(A ∩ B) = 15: n(A Δ B) = 15 + 25 - 2(15) = 10

Possible Values

The possible values of n(A Δ B) can be 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, and 40. These values range from 10 to 40 in steps of 2, which means there are a total of:

(40 - 10) / 2 + 1 = 16 possible values.

Thus, the number of possible values for the symmetric difference of sets A and B is 16.