Question

Let $R$ be a relation from a set $A$ to a set $B$ . The inverse relation from $B$ to $A,$ denoted by $R^{-1}$ , is the set of ordered pairs $\{(b, a) |(a, b) \in R\} .$ The complementary relation $\overline{R}$ is the set of ordered pairs $\{(a, b) |(a, b) \notin R\}$. Let $R$ be the relation $R=\{(a, b) | a<b\}$ on the set of integers. Find $\begin{array}{ll}{\text { a) } R^{-1}} & {\text { b) } \overline{R}}\end{array}$

Name: Problem 26, Chapter 9: Discrete Mathematics and its Applications
Uploaded: 2021-01-23T19:42:41-08:00
Duration: 1 min 18 s
Channel: Adam Dehollander
Description: Let R be a relation from a set A to a set B . The inverse relation from B to A, denoted by R^-1 , is the set of ordered pairs {(b, a) |(a, b) ∈R} . The complementary relation R is the set of ordered pairs {(a, b) |(a, b) ∉R}. Let R be the relation R={(a, b) | a<b} on the set of integers. Find a) R^-1 b) R

   Let $R$ be a relation from a set $A$ to a set $B$ . The inverse relation from $B$ to $A,$ denoted by $R^{-1}$ , is the set of ordered pairs $\{(b, a) |(a, b) \in R\} .$ The complementary relation $\overline{R}$ is the set of ordered pairs $\{(a, b) |(a, b) \notin R\}$.
Let $R$ be the relation $R=\{(a, b) | a<b\}$ on the set of integers. Find
$\begin{array}{ll}{\text { a) } R^{-1}} & {\text { b) } \overline{R}}\end{array}$

Discrete Mathematics and its Applications

Kenneth Rosen 8th Edition

Chapter 9, Problem 26

Instant Answer

Solved by Expert Adam Dehollander

01/23/2021

Step 1

Since $R$ is defined as the set of ordered pairs $(a, b)$ such that $a < b$, the inverse relation $R^{-1}$ is the set of ordered pairs $(b, a)$ such that $b < a$. This can be rewritten as $(a, b)$ such that $a > b$. Show more…

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Let $R$ be a relation from a set $A$ to a set $B$ . The inverse relation from $B$ to $A,$ denoted by $R^{-1}$ , is the set of ordered pairs $\{(b, a) |(a, b) \in R\} .$ The complementary relation $\overline{R}$ is the set of ordered pairs $\{(a, b) |(a, b) \notin R\}$. Let $R$ be the relation $R=\{(a, b) | a<b\}$ on the set of integers. Find $\begin{array}{ll}{\text { a) } R^{-1}} & {\text { b) } \overline{R}}\end{array}$

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Key Concepts

Inverse Relation

The inverse relation of a given relation R is formed by reversing the order of the elements in each ordered pair of R. That is, for each pair (a, b) in R, the pair (b, a) is in the inverse relation R?¹. This concept helps in analyzing how relationships behave when the direction of association is reversed.

Complementary Relation

The complementary relation of a relation R is defined as the set of all ordered pairs that do not belong to R. It essentially represents all pairs in the Cartesian product of the two sets that are not related by R, providing a way to consider what lies outside of the given relation.

Binary Relation

A binary relation is a collection of ordered pairs, each consisting of an element from one set and an element from another (or the same) set. These relations are fundamental in mathematics as they describe associations or connections between elements of sets.

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