Chapter 9: Q3RE (page 634)
Determine an example of a relation on the set \(\{ 1,2,3,4\} \) that is Reflexive, symmetric and transitive.
The transitive relation is defined.
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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.
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) \({R_3} \oplus {R_5}\).
(a)To find the number of relations on the set \(\{ a,b,c,d\} \).
(b)To find the number of relations on the set \(\{ a,b,c,d\} \) contain the pair \((a,a)\).
Which relations in Exercise 4 are irreflexive?
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 .
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) \({R_2} \oplus {R_4}\).
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