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Chapter 7: Problem 21
Let \(|A|=5\). (a) How many directed graphs can one construct on \(A\) ? (b) How many of the graphs in part (a) are actually undirected?
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For \(|A|=5\), how many relations \(\mathscr{\text { on }} A\) are there? How many of these relations are symmetric?
Let \(A=\\{1,2,3,4,5\\} \times\\{1,2,3,4,5\\}\), and define \(\Re\) on \(A\) by \(\left(x_{1}, y_{1}\right) \mathscr{R}\left(x_{2}, y_{2}\right)\) if \(x_{1}+y_{1}=\) \(x_{2}+y_{2}\). a) Verify that \(\mathscr{\text { is an equivalence relation on }} A\). b) Determine the equivalence classes \([(1,3)],[(2,4)]\), and \([(1,1)]\). c) Determine the partition of \(A\) induced by \(\Re\).
For \(A=\\{(-4,-20),(-3,-9),(-2,-4),(-1,-11),(-1,-3),(1,2),(1,5),(2,10)\), \((2,14),(3,6),(4,8),(4,12)\\}\), define the relation \(\Re\) on \(A\) by \((a, b) \Re R(c, d)\) if \(a d=b c\). a) Verify that \(\mathscr{\text { is an equivalence relation on }} A\). b) Find the equivalence classes \([(2,14)],[(-3,-9)]\), and \([(4,8)]\). c) How many cells are there in the partition of \(A\) induced by \(\mathscr{\text { ? }}\)
Let \(A\) be a set and \(I\) an index set where, for each \(i \in I, \mathscr{R}_{i}\) is a relation on \(A\). Prove or disprove each of the following. a) \(\bigcup_{i \in I} \mathscr{A}_{i}\) is reflexive on \(A\) if and only if each \(\mathscr{A}_{i}\) is reflexive on \(A\). b) \(\bigcap_{i \in I} \mathscr{R}_{i}\) is reflexive on \(A\) if and only if each \(\mathscr{R}\) is reflexive on \(A\).
How many (undirected) edges are there in the complete graphs \(K_{6}, K_{7}\), and \(K_{n}\), where \(n \in \mathbf{Z}^{+} ?\)
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