Chapter 9: Q30E (page 597)
How can the directed graph of a relation \(R\) on a finite set \(A\) be used to determine whether a relation is irreflexive?
The corresponded digraph must have no loop.
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Let \(R\) be the relation\(\{ (a,b)\mid a\;divides\;b\} \)on the set of integers. What is the symmetric closure of\(R\)?
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.
34. Find
(a) \({R_1} \cup {R_3}\).
(b) \({R_1} \cup {R_5}\).
(c) \({R_2} \cap {R_4}\).
(d) \({R_3} \cap {R_5}\).
(e) \({R_1} - {R_2}\).
(f) \({R_2} - {R_1}\).
(g) \({R_1} \oplus {R_3}\).
(h) \({R_2} \oplus {R_4}\).
To provethat \({R^n} = R\forall n \in {z^ + }\)when \(R\) is reflexive and transitive.
Show that if \(R\) and \(S\) are both \(n\)-ary relations, then
\({P_{{i_1},{i_2}, \ldots ,{i_m}}}(R \cup S) = {P_{{i_1},{i_2}, \ldots ,{i_m}}}(R) \cup {P_{{i_1},{i_2}, \ldots ,{i_m}}}(S)\).
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.
36. Find
(a) \({R_1}^\circ {R_1}\).
(b) \({R_1}^\circ {R_2}\).
(c) \({R_1}^\circ {R_3}\).
(d) \({R_1}^\circ {R_4}\).
(e) \({R_1}^\circ {R_5}\).
(f) \({R_1}^\circ {R_6}\).
(g) \({R_2}^\circ {R_3}\).
(h) \({R_3}^\circ {R_3}\).
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