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Chapter 7: Problem 32

Give an example of a relation on \(\\{a, b, c\\}\) that is: Antisymmetric, but not symmetric.

Short Answer

Expert verified
An example of a relation on \(\\{a, b, c\\}\) that is antisymmetric but not symmetric is \(R = \{ (a, b) \}\).

Step by step solution

01

Check Antisymmetry

To check for antisymmetry, we verify that if (x, y) is in R, then (y, x) is not in R for distinct elements x, y. We only have one pair, (a, b), and its reverse, (b, a), is not in the relation R. Hence, the relation R is antisymmetric.
02

Check Lack of Symmetry

To check that R is not symmetric, we check if there exists a pair (x, y) in R, such that the reverse pair (y, x) is not in R. We have the pair (a, b) in R, but its reverse (b, a) is not in R. So, the relation R is not symmetric. Since relation R satisfies both being antisymmetric and not symmetric, it is a valid example. Thus, an example of a relation on {a, b, c} that is antisymmetric but not symmetric is: \(R = \{ (a, b) \}\)

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Key Concepts

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

Antisymmetric Relation
An antisymmetric relation is a specific type of relation in set theory that has a unique property. For a relation \(R\) on a set \(S\), it is antisymmetric if for every pair of elements \(x\) and \(y\) in \(S\), whenever both \((x, y)\) and \((y, x)\) are in \(R\), it must be true that \(x = y\). In simpler words:
  • If \(x\) is related to \(y\), and \(y\) is related to \(x\), then \(x\) must be equal to \(y\).
  • There should not be two different elements related to each other in both directions unless they are the same.
Consider the example given: the relation \(R = \{(a, b)\}\) on the set \(\{a, b, c\}\). This is antisymmetric because there is only one pair \((a, b)\) and there isn't a pair \((b, a)\) in \(R\). Thus, the condition for antisymmetry is satisfied, as there are no two distinct elements that relate to each other mutually.
Symmetric Relation
A symmetric relation has a property that whenever one element is related to another, the second is related back to the first. Formally, for a relation \(R\) on a set \(S\), it is symmetric if:
  • For every \((x, y)\) in \(R\), the reverse pair \((y, x)\) also belongs to \(R\).
This means if \(a\) is related to \(b\), then \(b\) should also be related to \(a\).
In the given step-by-step breakdown, the relation \(R = \{(a, b)\}\) is checked for symmetry. Since \((a, b)\) is in \(R\) but \((b, a)\) is not, the relation is not symmetric. The absence of \((b, a)\) in \(R\) clearly breaks the condition needed for symmetry, showing how the relation \(R\) fails to be symmetric while ensuring it's antisymmetric.
Set of Elements
In set theory, a set is a collection of distinct objects or elements considered as a whole. Each element in a set is unique.
The set of elements plays a fundamental role in defining relations, because relations are defined over these sets.
  • For example, consider a set \(\{a, b, c\}\). This is a simple set with three distinct elements.
  • You can form a relation by considering various pairings of these elements, such as \((a, b)\), \((b, c)\), etc.
In the context of the exercise, we start with the set \(\{a, b, c\}\) and consider only the pair \((a, b)\) to define our relation \(R\). Sets and their pairings are the building blocks for exploring deeper concepts like symmetry and antisymmetry in relations. By understanding the individual elements and their possible pairings, we gain a clearer idea of how relations behave on these sets.

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