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Chapter 9: Problem 29

Find the smallest relation containing the relation {(1, 2), (1, 4), (3, 3), (4, 1)} that is a) reflexive and transitive. b) symmetric and transitive. c) reflexive, symmetric, and transitive.

Short Answer

Expert verified
Smallest reflexive and transitive relation: {(1,1), (1,2), (1,4), (2,2), (3,3), (4,4)}. Smallest symmetric and transitive relationship: {(1,2), (2,1), (3,3), (4,1), (4,2), (4,4)}. Smallest reflexive, symmetric, and transitive relationship: {(1,1), (1,2), (1,4), (2,2), (2,1), (3,3), (4,4), (4,1), (4,2)}.

Step by step solution

01

Title - Identify Reflexive Closure

A relation is reflexive if every element is related to itself. For the relation to be reflexive, add (1,1), (2,2), (3,3), and (4,4). Note that (3, 3) is already present.
02

Title - Identify Transitive Closure

A relation is transitive if whenever (a, b) and (b, c) are in the relation, (a, c) is also in the relation. For (1, 2) and (2, 2), add (1, 2). For (1, 4) and (4, 1), add (1, 1). Note (1, 1) already exists. The additional pairs are (1, 2) which is already included.
03

Title - Combine Reflexive and Transitive Closures

Combining both reflexive and transitive closures, the smallest reflexive and transitive relation includes: (1, 1), (1, 2), (1, 4), (2, 2), (3, 3), and (4, 4).
04

Title - Identify Symmetric Closure

A relation is symmetric if whenever (a, b) is in the relation, (b, a) is also in the relation. For (1, 2), add (2, 1). For (1, 4), add (4, 1). The pairs (1, 2) and (2, 1) ensure transitivity from previous steps.
05

Title - Combine Symmetric and Transitive Closures

Combining symmetric and transitive closures: for (4, 1) and (1, 2), add (4, 2). For (1, 4) and (4, 1), (1, 1) is already included. Continue to mitigate transitivity for additional pairs. The additional pair closing all solutions is: (1, 2), (2, 1), (3, 3), (4, 1), (4, 2), (4, 4).
06

Title - Combine Reflexive, Symmetric, and Transitive Closures - First Five Steps

Combining reflexive, symmetric, and transitive closures from identified steps: (1,1), (2,2), (3,3), (4,4), (1, 2), (2,1), (4,1), (1,4), (4,2). Ensure each added reflexive and transitive mitigation/preferences match.

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Key Concepts

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

Reflexive Closure
In discrete mathematics, a relation is considered reflexive if every element maps to itself. This means that for a relation to achieve its reflexive closure, we must include pairs where each element of the set is related to itself.

For instance, if you have a set containing elements {1, 2, 3, 4} and the relation {(1, 2), (1, 4), (3, 3), (4, 1)}, the reflexive closure requires us to add the pairs (1, 1), (2, 2), and (4, 4) since these self-relations are not already present. Note that (3, 3) is already included in the original relation.

Therefore, to make the initial set reflexive, we introduce these new pairs, yielding the extended relation:
  • (1, 1) - to relate '1' to itself
  • (2, 2) - to relate '2' to itself
  • (3, 3) - already present
  • (4, 4) - to relate '4' to itself
These additions ensure that every element in the set is connected to itself.
Transitive Closure
A relation is transitive if for any elements a, b, and c, whenever (a, b) and (b, c) are in the relation, the pair (a, c) must also be included.

To determine the transitive closure of a relation, it's essential to ensure that all transitive pairs are present. Taking the same relation {(1, 2), (1, 4), (3, 3), (4, 1)}, to find the transitive closure:
  • From (1,2) and (2,2), we would expect (1,2) - already included.
  • From (1,4) and (4,1), we would expect (1,1) - already included after reflexive.
Combining these pairs with existing ones, we derive the smallest relation possessing both reflexive and transitive properties.
Symmetric Closure
For a relation to be symmetric, if it includes a pair (a, b), it must also contain the pair (b, a). Symmetric closure ensures this symmetry.

Given the relation {(1, 2), (1, 4), (3, 3), (4, 1)}, we form its symmetric closure:
  • Since (1, 2) is in the relation, we add (2, 1).
  • Since (1, 4) is in the relation, we add (4, 1), already present.
By adding these pairs, the relation becomes symmetric. We now check for combined symmetry and transitivity to form closure.

For instance, ensuring transitivity with (4, 1) and (1, 2) should imply (4, 2). Adding such pairs ensures the relation is symmetric and maintains transitivity, making sure we cover (4, 2).

Combining symmetry with already established reflexive and transitive conditions gives a relations set that meets all specified criteria.

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Most popular questions from this chapter

Determine whether the relation \(R\) on the set of all real numbers is reflexive, symmetric, antisymmetric, and/or transitive, where \((x, y) \in R\) if and only if a) \(x+y=0\) b) \(x=\pm y\) c) \(x-y\) is a rational number d) \(x=2 y\) e) \(x y \geq 0\) f) \(x y=0\) g) \(x=1\) h) \(x=1\) or \(y=1\)

a) How many relations are there on the set \(\\{a, b, c, d\\} ?\) b) How many relations are there on the set \(\\{a, b, c, d\\}\) that contain the pair \((a, a) ?\)

Determine whether the relation \(R\) on the set of all people is reflexive, symmetric, antisymmetric, and/or transitive, where \((a, b) \in R\) if and only if a) \(a\) is taller than \(b\). b) \(a\) and \(b\) were born on the same day. c) \(a\) has the same first name as \(b\) . d) \(a\) and \(b\) have a common grandparent.

Represent each of these relations on \(\\{1,2,3,4\\}\) with a matrix (with the elements of this set listed in increasing order). a) \(\\{(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)\\}\) b) \(\\{(1,1),(1,4),(2,2),(3,3),(4,1)\\}\) c) \(\\{(1,2),(1,3),(1,4),(2,1),(2,3),(2,4),(3,1),(3,2)\) (3, \(4 ),(4,1),(4,2),(4,3) \\}\) d) \(\\{(2,4),(3,1),(3,2),(3,4)\\}\)

Let \(R\) be a relation on a set \(A\) with \(n\) elements. If there are \(k\) nonzero entries in \(\mathbf{M}_{R},\) the matrix representing \(R,\) how many nonzero entries are there in \(\mathbf{M}_{R^{-1}},\) the matrix representing \(R^{-1}\) , the inverse of \(R ?\)
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