Chapter 9: Q16E (page 597)
To calculate the number of non-zero entries in the matrix \({M_{{R^{ - 1}}}}\).
The number of non-zero entries of \({M_{{R^{ - 1}}}}\) is also \(k\).
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Which relations in Exercise 5 are asymmetric?
Exercises 34–37 deal with these relations on the set of real numbers:
\({R_1} = \left\{ {\left( {a,\;b} \right) \in {R^2}|a > b} \right\},\)the “greater than” relation,
\({R_2} = \left\{ {\left( {a,\;b} \right) \in {R^2}|a \ge b} \right\},\)the “greater than or equal to” relation,
\({R_3} = \left\{ {\left( {a,\;b} \right) \in {R^2}|a < b} \right\},\)the “less than” relation,
\({R_4} = \left\{ {\left( {a,\;b} \right) \in {R^2}|a \le b} \right\},\)the “less than or equal to” relation,
\({R_5} = \left\{ {\left( {a,\;b} \right) \in {R^2}|a = b} \right\},\)the “equal to” relation,
\({R_6} = \left\{ {\left( {a,\;b} \right) \in {R^2}|a \ne b} \right\},\)the “unequal to” relation.
35. Find
(a) \({R_2} \cup {R_4}\).
(b) \({R_3} \cup {R_6}\).
(c) \({R_3} \cap {R_6}\).
(d) \({R_4} \cap {R_6}\).
(e) \({R_3} - {R_6}\).
(f) \({R_6} - {R_3}\).
(g) \({R_2} \oplus {R_6}\).
(h) \({R_3} \oplus {R_5}\).
To prove that the relation \(R\) on a set \(A\) is symmetric if and only if \(R = {R^{ - 1}}\) where \({R^{ - 1}}\) is the inverse relation.
Let \(R\) be the relation \(\{ (a,b)\mid a \ne b\} \) on the set of integers. What is the reflexive closure of \(R\)?
Whether there is a path in the directed graph in Exercise 16 beginning at the first vertex given and ending at the second vertex given.
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