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Is my set of related pairs correct for this problem? $$\{(2,1), (2,2), (3,1), (3,2), (3,3), (4,1), (4,2), (4,3), (4,4)\}$$

Suppose that $\,A = \{1,2,3,4\}\,$ and $\,B = \{1, 2, 3\}.$

Let $R$ be the relation from $A$ to $B$ containing $(a,b)$ if $\,a \in A$, $b \in B$, and $\;a^2 \ge 2^b$.

What is the matrix representing $R$? (Assume the ordering of elements is the same as the increasing numerical order.)

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You are correct in the set you've written to represent the relation, except we cannot have $(4, {\bf 4})$ in the relation, since ${\bf 4} \notin B$

Now, with respect to the matrix representation, it's an incident matrix we want, but we need to consider $12 = 4 \times 3$ pairs $(a, b) \in A\times B$.

We can do this by using a $4\times 3$ matrix, each row corresponding to one element in $A $, each column corresponding to each of the elements of $B$. Then the $m_{i,j}$ entry will be a $1$ if $(a_i, b_j)$ is one pair in the set of relations, and $0$ otherwise.

$$M = \begin{pmatrix} 0 & 0 & 0\\ 1 & 1 & 0 \\ 1 & 1 & 1 \\ 1 & 1 & 1\end{pmatrix}$$

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  • $\begingroup$ Oh I'm such a doofus! Obvious things sometimes evade me. Thank you! $\endgroup$ Jun 23, 2013 at 4:02
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    $\begingroup$ You're welcome! I was puzzling over why you chose a 4x4 matrix, when there were only three elements in B. Then I saw you mistakenly included the pair (4, 4). We all goof up sometimes. $\endgroup$
    – amWhy
    Jun 23, 2013 at 4:04
  • $\begingroup$ @amWhy: Great feedback +1 $\endgroup$
    – Amzoti
    Jun 23, 2013 at 5:08

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