Chapter 9: Q1E (page 615)
Which of these relations on \(\{ 0,1,2,3\} \) are equivalence relations? Determine the properties of an equivalence relation that the others lack.
\(\{ (0,0),(0,1),(0,2),(1,0),(1,1),(1,2),(2,0),(2,2),(3,3)\} \) is not an equivalence relation.
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To prove the closure with respect to the property. ofthe relation \(R = \{ (0,0),(0,1),(1,1),(2,2)\} \) on the set \(\{ 0,1,2\} \) does not exist if . is the property" has an odd number of elements."
Which relations in Exercise 4 are irreflexive?
Which 4-tuples are in the relation \(\{ (a,b,c,d)\mid a,b,c\), and \(d\) are positive integers with \(abcd = 6\} \) ?
Show that if \(R\) and \(S\) are both \(n\)-ary relations, then
\({P_{{i_1},{i_2}, \ldots ,{i_m}}}(R \cup S) = {P_{{i_1},{i_2}, \ldots ,{i_m}}}(R) \cup {P_{{i_1},{i_2}, \ldots ,{i_m}}}(S)\).
How many transitive relations are there on a set with \(n\) elements if
a) \(n = 1\) ?
b) \(n = 2\) ?
c) \(n = 3\) ?
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