Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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Question

Please do Exercise 17.2.18 part A,B,C,and D. Please show step by step and explain

**Exercise 17.2.18**: Find binary relations on \(\{1, 2, 3\}\) that meet each of the following conditions. Express each relation as a set of ordered pairs, and draw the corresponding digraph. (Note: each part can have more than one answer, but you only need to find one.) (a) Symmetric, but neither reflexive nor transitive. (b) Reflexive, but neither symmetric nor transitive. (c) Transitive and symmetric, but not reflexive. (d) Neither reflexive, nor symmetric, nor transitive. **Explanation:** A binary relation from a set \(A\) to itself is a subset of \(A \times A\), the set of all ordered pairs of elements from \(A\). Below is a brief description of what each property means: - **Symmetric**: If \((a, b)\) is in the relation, then \((b, a)\) must also be in the relation. - **Reflexive**: Every element is related to itself; i.e., for each element \(a\) in the set, \((a, a)\) is in the relation. - **Transitive**: If \((a, b)\) and \((b, c)\) are in the relation, then \((a, c)\) must be in the relation. For each condition, a corresponding digraph (directed graph) can be drawn by representing each element as a vertex and each ordered pair \((a, b)\) as a directed edge from vertex \(a\) to vertex \(b\).
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Transcribed Image Text:**Exercise 17.2.18**: Find binary relations on \(\{1, 2, 3\}\) that meet each of the following conditions. Express each relation as a set of ordered pairs, and draw the corresponding digraph. (Note: each part can have more than one answer, but you only need to find one.) (a) Symmetric, but neither reflexive nor transitive. (b) Reflexive, but neither symmetric nor transitive. (c) Transitive and symmetric, but not reflexive. (d) Neither reflexive, nor symmetric, nor transitive. **Explanation:** A binary relation from a set \(A\) to itself is a subset of \(A \times A\), the set of all ordered pairs of elements from \(A\). Below is a brief description of what each property means: - **Symmetric**: If \((a, b)\) is in the relation, then \((b, a)\) must also be in the relation. - **Reflexive**: Every element is related to itself; i.e., for each element \(a\) in the set, \((a, a)\) is in the relation. - **Transitive**: If \((a, b)\) and \((b, c)\) are in the relation, then \((a, c)\) must be in the relation. For each condition, a corresponding digraph (directed graph) can be drawn by representing each element as a vertex and each ordered pair \((a, b)\) as a directed edge from vertex \(a\) to vertex \(b\).
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