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Applied Mathematics · Class 11 Commerce

Ch 4Relations — Class 11 Applied Mathematics, concept-first.

This concept map shows how the chapter fits together. A relation is built from ordered pairs and the Cartesian product; it has a domain and a range; and relations come in several types — empty, universal, reflexive, symmetric and transitive (a relation that is all three is an equivalence relation), leading on to functi…

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Introduction

You've been solving problems your whole life. When you see a new type of question, you first figure out what it's asking, then decide which tools to use, then carry out the steps.

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Chapter contents

The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.

Concept Map

This concept map shows how the chapter fits together. A relation is built from ordered pairs and the Cartesian product; it has a domain and a range; and relations come in several types — empty, univer…

4.1

Introduction

We already know how to pair numbers and objects in everyday life — a student with their roll number, a city with its pin code.

4.2

Ordered Pair

An ordered pair is a pair of objects written in a fixed order — usually as — where is called the first component and the second.

+Worked Examplesi2 questions
  1. Example 1If the ordered pairs $(2x, y - 3)$ and $(2, 1)$ are equal, then find $x$ and $y$.Free
  2. Example 2Find 'x' and 'y' if $(x^2 - 4x,\ y^2 - y) = (-4, 6)$Preview
4.3

Cartesian Product of Two Sets

The Cartesian product is the bridge that turns two separate sets into a single set of ordered pairs, where the first element always comes from the first set and the second from the second.

+Worked Examplesi4 questions
  1. Example 3Let $A = \{a, b, c\}$, $B = \{1, 2\}$. Find $A \times B$ and $B \times A$. Is $A \times B = B \times A$?Free
  2. Example 4Let $A = \{2, 3\}$ and $B = \{4, 5\}$. Find: 1. $A \times B$ 2. $B \times A$ 3. $n(A \times B)$ 4. number of subsets of $A \times B$Free
  3. Example 5If $A \times B = \{(a, x), (a, y), (b, x), (b, y)\}$, then find $A$ and $B$.Preview
  4. Example 6If $(-1, 1), (2, 3), (1, 0), (2, 1)$ are some of the elements of $A \times B$, then find $A$ and $B$. Also find the remaining elements of $A…Preview
4.4

Relations

A relation is simply a connection between two sets of things. For instance, if you have a set of students and a set of subjects, the idea "is enrolled in" pairs each student with the subjects they stu…

+Worked Examplesi2 questions
  1. Example 7Let $A = \{1, 2, 3, 4, 5, 6\}$. Define a relation $R$ from $A$ to $A$ by: $R = \{(x, y) : x > y\}$ 1. Write the given relation in tabular or…Free
  2. Example 8Let $A$ be a non-empty set with $n(A) = 3$. Find the number of relations from set $A$ to itself.Preview
4.5

Types of Relations

13 Q

Relations are the backbone of how we connect elements from one set to another, but not all connections are created equal.

+Worked Examplesi8 questions
  1. Example 9Let $A$ be the set of all natural numbers. Define $R = \{(x, y);\ \frac{x}{y} < 0;\ x, y \in A\}$.Free
  2. Example 10Let $A = \{1, 2, 3, 4, 5\}$. Define $R = \{(x, y) : x + y \in N;\ x, y \in A\}$.Free
  3. Example 11Let $T$ be the set of all triangles in a plane with $R$ a relation in $T$ given by $R = \{(T_1, T_2) : T_1 \sim T_2\}$. Show that $R$ is an…Free
  4. Example 12Let $S$ be any non-empty set and $R$ be a relation defined on the power set of $S$, i.e., on $P(S)$ by $A\,R\,B$ iff $A \subset B$ for all $…Preview
  5. Example 13Show that the relation $R$ in the set $Z$ of integers given by $R = \{(a, b) : 3 \text{ divides } a - b\}$ is an equivalence relation.Preview
  6. Example 14Examine each of the following relations given below and state, giving reasons, whether it is a function or not? (i) $R = \{(2, 1), (3, 2), (…Preview
  7. Example 15Consider the following arrow diagrams depicting relations from set $A$ to set $B$. Which amongst them are functions? Give reasons. (i) $A =…Preview
  8. Example 16Find the domain and range of each of the following real functions: (i) $f(x) = x^2$ (ii) $f(x) = \frac{1}{x}$Preview
+Exercise 4.1i5 questions
  1. Q1Determine whether each of the following relations are reflexive, symmetric and transitive. (i) Relation $R$ in a set $S = \{1, 2, 3, 4, 5\}$…Free
  2. Q2Show that the relation $R$ in the set $\mathbb{R}$ of real numbers, defined as $R = \{(a, b) : a < b^2\}$ is neither reflexive nor symmetric…Free
  3. Q3Show that the relation $R$ in the set $Z$ of integers given by $R = \{(a, b) : 2 \text{ divides } a - b\}$ is an equivalence relation.Preview
  4. Q4If $X = \{1, 2, 3, 4\}$, give an example on $X$ which is (i) reflexive and symmetric but not transitive. (ii) symmetric and transitive but n…Preview
  5. Q5If $R_1$ and $R_2$ are equivalence relations in set $A$, show that $R_1 \cap R_2$ is also an equivalence relation.Preview