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Chapter 9: Problem 56

Show that the relation \(R\) on a set \(A\) is reflexive if and only if the complementary relation \(\overline{R}\) is irreflexive.

Short Answer

Expert verified
A relation \(R\) is reflexive if and only if \(\bar{R}\) is irreflexive because \( \forall a \, \text{in} \, A, (a, a) \, \text{is in} \, R \) implies \( (a, a) \, \text{is not in} \, \bar{R}\) and vice versa.

Step by step solution

01

Understanding Reflexivity

A relation \(R\) on a set \(A\) is reflexive if every element in \(A\) is related to itself. Formally, \(R\) is reflexive if \( \forall a \, \text{in} \, A, (a, a) \, \text{is in} \, R\).
02

Understanding Irreflexivity

A relation \(\bar{R}\) (complementary of \(R\)) on a set \(A\) is irreflexive if no element in \(A\) is related to itself. Formally, \(\bar{R}\) is irreflexive if \( \forall a \, \text{in} \, A, (a, a) \, \text{is not in} \, \bar{R}\).
03

Complementary Relation

The complementary relation \(\bar{R}\) consists of all the pairings in \(A \times A\) that are not in \(R\). Therefore, \((a, b) \, \text{is in} \, \bar{R}\) if and only if \((a, b) \, \text{is not in} \, R\).
04

Prove Reflexivity implies Irreflexivity of the Complement

Assume \(R\) is reflexive. Then, by definition, \( \forall a \, \text{in} \, A, (a, a) \, \text{is in} \, R\). Consequently, \( (a, a) \, \text{is not in} \, \bar{R}\) for all \(a\). This implies that \(\bar{R}\) is irreflexive.
05

Prove Irreflexivity of the Complement implies Reflexivity

Assume \(\bar{R}\) is irreflexive. Then, by definition, \( \forall a \, \text{in} \, A, (a, a) \, \text{is not in} \, \bar{R}\). This means that \( (a, a) \, \text{is in} \, R\) for all \(a\). Hence, \(R\) is reflexive.
06

Conclusion

Therefore, we have shown that a relation \(R\) is reflexive if and only if its complementary relation \(\bar{R}\) is irreflexive.

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Key Concepts

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

Relations in Discrete Mathematics
In discrete mathematics, relations are a way of describing a relationship between elements of a set. When we talk about a relation on a set \(A\), it typically refers to a subset of the Cartesian product \(A \times A\). This means a relation \(R\) can be understood as any collection of ordered pairs \((a, b)\) where both \(a\) and \(b\) belong to \(A\).

Types of relations include:
  • Reflexive Relation: Every element is related to itself. Mathematically, \(R\) is reflexive if \(\forall a \in A, (a, a) \in R\).
  • Irreflexive Relation: No element is related to itself. Mathematically, \(\forall a \in A, (a, a) otin R\).
  • Symmetric Relation: If \(a\) is related to \(b\), then \(b\) is related to \(a\). Mathematically, \(\forall a, b \in A, (a, b) \in R \Rightarrow (b, a) \in R\).
  • Transitive Relation: If \(a\) is related to \(b\) and \(b\) is related to \(c\), then \(a\) is related to \(c\). Mathematically, \(\forall a, b, c \in A, (a, b) \in R \wedge (b, c) \in R \Rightarrow (a, c) \in R\).
Understanding these concepts is vital in recognizing patterns in data, structures in mathematics, and simplifications in complex problems.
Complementary Relation
The complementary relation \( \overline{R} \) of a given relation \( R \) on a set \( A \) consists of all pairs \((a, b)\) in \(A \times A\) that are not in \(R\). Essentially, if \( (a, b) \in \overline{R} \), then \( (a, b) otin R \), and vice versa.

Here's a deeper look:
  • Complement: The concept of a complement is akin to taking everything that is not in the original set. In this context, it means every ordered pair not present in \( R \).
  • Example: If \( A = \{1, 2\} \) and \( R = \{(1, 1), (2, 2)\} \), then the complement \( \overline{R} \) will be \( \{(1, 2), (2, 1)\} \).
For relations, understanding complements helps in proving properties about the relations themselves, as seen in showing that a reflexive relation \( R \) implies its complement \( \overline{R} \) is irreflexive.
Proof Techniques
Proving properties about relations often involves clear logical steps. Let's break down the key techniques used in proofs concerning reflexivity and irreflexivity.

Understanding Definitions: First, it's crucial to understand the definitions involved. For reflexive relations, we need \( (a, a) \in R \) for all \( a \in A \). For irreflexive relations, we need \( (a, a) otin \overline{R} \) for all \( a \in A \).

Direct Proof Method: Often we use direct proof where we assume one property and then show another:
  • To prove that \( R \) being reflexive implies that \( \overline{R} \) is irreflexive, start by assuming \( R \) is reflexive.
  • We then show that \( (a, a) otin \overline{R} \), meaning \( \overline{R} \) is irreflexive.

Contrapositive Proof Method: Sometimes proving a contrapositive can simplify the effort:
  • Instead of directly proving \( P \Rightarrow Q \), we prove \( eg Q \Rightarrow eg P \).
  • If \( \overline{R} \) was not irreflexive, it would mean \( R \) is not reflexive.
Clear and methodical proof helps in understanding and verifying mathematical properties.

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Most popular questions from this chapter

Determine whether these posets are lattices. a) \((\\{1,3,6,9,12\\}, |)\) b) \((\\{1,5,25,125\\}, |)\) c) \((\mathbf{Z}, \geq)\) d) \((P(S), \supseteq),\) where \(P(S)\) is the power set of a set \(S\)

A relation \(R\) on the set \(A\) is irreflexive if for every \(a \in A,(a, a) \notin R .\) That is, \(R\) is irreflexive if no element in \(A\) is related to itself. Can a relation on a set be neither reflexive nor irreflexive?

Let \(R_{1}=\\{(1,2),(2,3),(3,4)\\}\) and \(R_{2}=\\{(1,1),(1,2)\) \((2,1),(2,2),(2,3),(3,1),(3,2),(3,3),(3,4) \\}\) be relations from \(\\{1,2,3\\}\) to \(\\{1,2,3,4\\} .\) Find $$ \begin{array}{ll}{\text { a) } R_{1} \cup R_{2}} & {\text { b) } R_{1} \cap R_{2}} \\ {\text { c) } R_{1}-R_{2}} & {\text { d) } R_{2}-R_{1}}\end{array} $$

Give an example of an infinite lattice with a) neither a least nor a greatest element. b) a least but not a greatest element. c) a greatest but not a least element. d) both a least and a greatest element.

A relation \(R\) on the set \(A\) is irreflexive if for every \(a \in A,(a, a) \notin R .\) That is, \(R\) is irreflexive if no element in \(A\) is related to itself. Use quantifiers to express what it means for a relation to be irreflexive.
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