Chapter 9: Q6RE (page 634)
How to use a zero-one matrix to represent a relation to determine whether the relation is reflexive, symmetric or antisymmetric?
Answer is detailed below.
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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 determine for each of these relations on the set decide whether it is reflexive, whether it is symmetric, whether it is anti symmetric, and whether it is transitive .
What do you obtain when you apply the selection operator \({s_C}\), where \(C\) is the condition (Project \( = 2\) ) \( \wedge \) (Quantity \( \ge 50\) ), to the database in Table 10 ?
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) \({R_3}^\circ {R_3}\).
Which relations in Exercise 4 are irreflexive?
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