Question

P and Q are two sets such that \( n(P) = 3 \) and \( n(Q) = 4 \). A relation from \( P \) into \( Q \) is selected at random. What is the probability that the relation is not a function?

• A. \( \frac{63}{64} \)
• B. \( \frac{31}{32} \)
• C. \( \frac{15}{16} \)
• D. \( \frac{127}{128} \)

Hint:

The number of relations from set \( P \) to set \( Q \) is \( n(Q)^{n(P)} \), and the number of functions is \( n(Q)^{n(P)} \).

Answer 1

The Correct Option is A

The total number of relations from \( P \) to \( Q \) is given by: \[ n(Q)^n(P) = 4^3 = 64. \] For the relation to be a function, each element of \( P \) must be related to exactly one element of \( Q \). The number of functions is: \[ n(Q)^{n(P)} = 4^3 = 64. \] Thus, the number of non-functions is: \[ \text{Non-functions} = 4^3 - 4^3 = 63. \] Therefore, the probability that the relation is not a function is: \[ P(\text{not a function}) = \frac{63}{64}. \]