Chapter 9: Q62E (page 618)
Determine the number of different equivalence relations on a set with four elements by listing them.
There are \(15\) distinct equivalence relations on \(S\), as listed below.
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To find the ordered pairs in \({R^3}\) relation.
To determine Inverse relation for the given relation.
Use quantifiers to express what it means for a relation to be irreflexive.
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) \({R_2} \oplus {R_4}\).
Show that if \({C_1}\) and \({C_2}\) are conditions that elements of the \(n\)-ary relation \(R\) may satisfy, then \({s_{{C_1} \wedge {C_2}}}(R) = {s_{{C_1}}}\left( {{s_{{C_2}}}(R)} \right)\).
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