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Chapter 10: Problem 3

A number of binary relations are defined on the set \(A=\\{0,1,2,3\\}\). For each relation: a. Draw the directed graph. b. Determine whether the relation is reflexive. c. Determine whether the relation is symmetric. d. Determine whether the relation is transitive. Give a counterexample in each case in which the relation does not satisfy one of the properties. $$R_{3}=\\{(2,3),(3,2)\\}$$

Short Answer

Expert verified
The relation \(R_{3}=\{(2,3),(3,2)\}\) has the following properties: a. The directed graph consists of nodes 0, 1, 2, and 3 with edges (2,3) and (3,2). b. The relation is not reflexive. Counterexample: (0,0) is not in \(R_{3}\). c. The relation is symmetric. d. The relation is transitive.

Step by step solution

01

Draw the directed graph

To draw the directed graph, represent each element of the set A as a node and the pairs in the relation as directed edges between these nodes. 1. Draw 4 nodes labeled 0, 1, 2, and 3 for each element of set A. 2. Draw a directed edge from node 2 to node 3 for the pair (2,3). 3. Draw a directed edge from node 3 to node 2 for the pair (3,2). The directed graph for the relation \(R_{3}\) is complete.
02

Check for reflexivity

A relation is reflexive if \((a,a) \in R\) for all elements \(a\) in set \(A\). In this case, since no pair of the form \((a,a)\) is present in \(R_{3}\), the relation is not reflexive. A counterexample to the reflexivity of this relation is (0,0) which is not in \(R_{3}\).
03

Check for symmetry

A relation is symmetric if for every pair \((a,b) \in R\), the pair \((b,a)\) is also in \(R\). In the given relation \(R_{3}\), when \((2,3) \in R_{3}\) its symmetric pair, \((3,2)\) is also in \(R_{3}\). Therefore, the relation \(R_{3}\) is symmetric.
04

Check for transitivity

A relation is transitive if for every pair \((a,b) \in R\) and \((b,c) \in R\), the pair \((a,c)\) is also in \(R\). In this case, since there are only two pairs, neither of which share a common element that can be used to create a new pair, there is no possibility for the presence of a transitive relation. Given that there is no possibility of creating a new pair (a, c) from existing pairs, the relation \(R_{3}\) is transitive.
05

Summary of properties

We can summarize the properties of the relation \(R_{3}=\{(2,3),(3,2)\}\) as follows: a. The directed graph is drawn with nodes (0, 1, 2, 3) and edges ((2,3), (3,2)). b. The relation is not reflexive (counterexample: (0,0)). c. The relation is symmetric. d. The relation is transitive.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Directed Graph
A directed graph is a powerful way to visualize relationships in mathematics, especially binary relations. Imagine each element of a set as a point or 'node'. Then, if two elements are related, we draw an arrow or 'directed edge' from one node to the other.

For example, with the set A = {0, 1, 2, 3} and the binary relation R3 = {(2,3), (3,2)}, we begin by plotting four nodes for each element of A. Then, we draw one arrow from node 2 to 3 to represent the pair (2,3), and another arrow from 3 to 2 for the pair (3,2). This visual illustration helps to clarify the connections between elements in a simple, and comprehensible way, much like a road map shows the direction of travel between cities.
Reflexive Relation
When we talk about reflexive relations, we're referring to a scenario where every element in a set relates to itself. It's like saying every person is his or her own sibling. In formal terms, for a set A, a relation R on A is reflexive if for every a in A, the pair (a, a) is in R.

For set A = {0, 1, 2, 3} and relation R3, the absence of pairs like (0,0), (1,1), (2,2), and (3,3) means it鈥檚 not reflexive. An example to show R3 is not reflexive is the missing pair (0,0). Every element needs to 'self-loop' in this type of relation.
Symmetric Relation
Think of symmetric relations as friendly handshakes: if person A shakes hands with person B, then person B should also be willing to shake hands with person A. In our set A, the relation R3 holds this symmetry - since (2,3) is in R3, and the reverse, (3,2), is also in R3. So, if any pair (a, b) is in the relation R, and you can find its flipside (b, a), then you know your relation has the symmetry of a mirror.
Transitive Relation
Transitive relations operate on a 'if...then' principle similar to dominos falling in line. For a transitive relation, if (a, b) and (b, c) are present, then this implies that (a, c) must also be a part of the relation. In the case of R3, there seems to be a missing link - we don鈥檛 have a third pair, like (2, c) or (a, 3), to complete a chain. Thus, without a third domino to fall, we cannot say that the relation fails to be transitive. It simply doesn't have the additional connections to test this property properly.

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Most popular questions from this chapter

Determine whether the given binary relation is reflexive, symmetric, transitive, or none of these. Justify your answers. \(D\) is the "divides" relation on Z: For all integers \(m\) and \(n, m D n \Leftrightarrow m \mid n\).

Determine whether the given binary relation is reflexive, symmetric, transitive, or none of these. Justify your answers. Let \(A\) be the set of all lines in the plane. A binary relation \(R\) is defined on \(A\) as follows: For all \(l_{1}\) and \(l_{2}\) in \(A, l_{1} R l_{2} \Leftrightarrow l_{1}\) is perpendicular to \(l_{2}\).

Determine whether the given binary relation is reflexive, symmetric, transitive, or none of these. Justify your answers. Let \(A\) be the "punctured plane"; that is, \(A\) is the set of all points in the Cartesian plane except the origin \((0,0)\). A binary relation \(R\) is defined on \(A\) as follows: For all \(p_{1}\) and \(p_{2}\) in \(A, p_{1} R p_{2} \Leftrightarrow p_{1}\) and \(p_{2}\) lie on the same half line emanating from the origin.

a. Find all binary relations from \(\\{0,1\\}\) to \(\\{1\\}\). b. Find all functions from \(\\{0,1\\}\) to \(\\{1\\}\). c. What fraction of the binary relations from \(\\{0,1\\}\) to \(\\{1\\}\) are functions?

Let \(P\) be the set of all people in the world and define a relation \(R\) on \(P\) as follows: For all \(x, y \in P\), \(x R y \quad \Leftrightarrow \quad x\) is no older then \(y\). Is \(R\) antisymmetric? Prove or give a counterexample.
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