Chapter 9: Q14E (page 630)
To find the pair of elements \((7,7)\) are comparable in the poset \(\left( {{z^ + },1} \right)\).
The pair of elements \((7,7)\) are comparable in the poset \(\left( {{z^ + },1} \right)\).
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How many different relations are there from a set with elements to a set with elements?
Find the error in the "proof" of the following "theorem."
"Theorem": Let \(R\) be a relation on a set \(A\) that is symmetric and transitive. Then \(R\) is reflexive.
"Proof": Let \(a \in A\). Take an element \(b \in A\) such that \((a,b) \in R\). Because \(R\) is symmetric, we also have \((b,a) \in R\). Now using the transitive property, we can conclude that \((a,a) \in R\) because \((a,b) \in R\)and \((b,a) \in R\).
(a) To find Relation \({R_1} \cup {R_2}\).
(b) To find Relation \({R_1} \cap {R_2}\).
(c) To find Relation \({R_1} - {R_2}\).
(d) To find Relation \({R_2} - {R_1}\).
(e) To find Relation \({R_1} \oplus {R_2}\).
To prove that the relation \(R\) on set \(A\) is anti-symmetric, if and only if \(R \cap {R^{ - 1}}\) is a subset of the diagonal relation \(\Delta = \{ (a,a)\mid a \in A\} \)
Adapt Algorithm 1 to find the reflexive closure of the transitive closure of a relation on a set with \(n\) elements.
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