Discussion

Let A and B be two finite sets. If \(n(A) = 7\) and the number of relations from A into B is 128, then \(n(B) =\)

Answer: D

\(n(B) =\) 1.

A relation from \(A\) to \(B\) is any subset of \(A \times B\), so the number of relations is \(2^{n(A)n(B)}\):

\[ \begin{aligned} 2^{7\,n(B)} &= 128 = 2^{7} \\ 7\,n(B) \\ &= 7 \\ n(B) \\ &= 1 \end{aligned} \]

With \(n(B) = 0\) there would be only \(2^{0} = 1\) relation (the empty one), and \(n(B) = 2, 3, 4\) would give \(2^{14}, 2^{21}, 2^{28}\) relations.

In general, with \(m\) and \(n\) elements there are \(mn\) ordered pairs and so \(2^{mn}\) relations.

No comment is present. Be the first to comment.

Loading…