Points to Remember: Relation and Function | Class 10th Mathematics

Points to Remember: Relation and Function

Key Concepts

• The Cartesian Product of \(A\) with \(B\) is defined as \(A \times B = \{(a, b) \mid a \in A, b \in B\}\).
• A relation R from \(A\) to \(B\) is always a subset of \(A \times B\). That is, \(R \subseteq A \times B\).
• A relation R from \(X\) to \(Y\) is a function if for every \(x \in X\) there exists only one \(y \in Y\).
• A function can be represented by:
  • (i) an arrow diagram
  • (ii) a tabular form
  • (iii) a set of ordered pairs
  • (iv) a graphical form
• Some types of functions:
  • (i) One-one function
  • (ii) Onto function
  • (iii) Many-one function
  • (iv) Into function
• Identity function: \(f(x) = x\)
• Reciprocal function: \(f(x) = \frac{1}{x}\)
• Constant function: \(f(x) = c\)
• Linear function: \(f(x) = ax + b, a \neq 0\)
• Quadratic function: \(f(x) = ax^2 + bx + c, a \neq 0\)
• Cubic function: \(f(x) = ax^3 + bx^2 + cx + d, a \neq 0\)
• For three non-empty sets \(A\), \(B\) and \(C\), if \(f : A \to B\) and \(g : B \to C\) are two functions, then the composition of \(f\) and \(g\) is a function \(g \circ f : A \to C\) will be defined as \((g \circ f)(x) = g(f(x))\) for all \(x \in A\).
• If \(f\) and \(g\) are any two functions, then in general, \(f \circ g \neq g \circ f\).
• If \(f\), \(g\) and \(h\) are any three functions, then \(f \circ (g \circ h) = (f \circ g) \circ h\).

Geometrical Understanding of Cartesian Product of Three Sets | 10th Maths

Cartesian Product of three Set - Illustration for Geometrical understanding

Cartesian Product of three Sets

If A, B, C are three non-empty sets then the cartesian product of three sets is the set of all possible ordered triplets given by

$$ A \times B \times C = \{ (a,b,c) \mid a \in A, b \in B, c \in C \} $$

Illustration for Geometrical understanding of cartesian product of two and three sets

Let's define our sets:

Set A {0, 1}

Set B {0, 1}

Set C {0, 1}

Step 1: Find the Cartesian Product of A and B (A × B)

{0, 1} × {0, 1}

Result (A × B): {(0, 0), (0, 1), (1, 0), (1, 1)}

Representing A×B in the xy - plane we get a picture shown in Fig. 1.5.

(A×B)×C = {(0, 0),(0,1),(1, 0),(1,1)} × {0,1}

= {(0, 0, 0),(0, 0,1),(0,1, 0),(0,1,1),(1, 0, 0),(1, 0,1)(1,1, 0),(1,1,1)}

Representing A×B ×C in the xyz - plane we get a picture as shown in Fig. 1.6

Thus, A×B represent vertices of a square in two dimensions and A×B ×C represent vertices of a cube in three dimensions.

NOTES

In general, cartesian product of two non-empty sets provides a shape in two dimensions and cartesian product of three non-empty sets provide an object in three dimensions.

Cartesian Product: Definition, Illustrations, and Solved Examples for Class 10 Maths`

Cartesian Product - Definition, Illustration, Example, Solution

Cartesian Product

Illustration 1

Let us consider the following two sets.

A is the set of 3 vegetables and B is the set of 4 fruits. That is,

A = {carrot, brinjal, ladies finger} and B = {apple, orange, grapes, strawberry}

What are the possible ways of choosing a vegetable with a fruit? (Fig.1.2)

Fig. 1.2

We can select them in 12 distinct pairs as given below.

(c, a), (c, o), (c, g), (c, s), (b, a), (b, o), (b, g), (b, s), (l, a), (l, o), (l,g), (l, s)

This collection represents the cartesian product of the set of vegetables and set of fruits.

Definition

If A and B are two non-empty sets, then the set of all ordered pairs (a, b) such that \( a \in A \), \( b \in B \) is called the Cartesian Product of A and B, and is denoted by \( A \times B \). Thus, \( A \times B = \{(a,b) \mid a \in A, b \in B\} \).

Note

• \( A \times B \) is the set of all possible ordered pairs between the elements of A and B such that the first coordinate is an element of A and the second coordinate is an element of B.
• \( B \times A \) is the set of all possible ordered pairs between the elements of A and B such that the first coordinate is an element of B and the second coordinate is an element of A.
• If \( a = b \), then \( (a, b) = (b, a) \).
• The “cartesian product” is also referred as “cross product”.

Illustration 2

Let A = {1, 2, 3} and B = {a, b}. Write \( A \times B \) and \( B \times A \) ?

\( A \times B = \{1,2,3\} \times \{a,b\} = \{(1, a ),(1, b ),(2, a ),(2, b ),(3, a ),(3, b )\} \) (as shown in Fig.1.3)

\( B \times A = \{a,b\} \times \{1,2,3\} = \{(a,1), (a,2), (a,3),(b,1), (b,2), (b,3)\} \) (as shown in Fig.1.3)

Fig. 1.3

Recall of standard infinite sets

Natural Numbers N = {1, 2, 3, 4…}

Whole Numbers W = {0,1,2,3, ...}

Integers Z ={..., –2,–1,0,1,2, ...}

Rational Numbers \( \mathbf{Q} = \{ \frac{p}{q} \mid p, q \in \mathbf{Z}, q \neq 0 \} \)

Real Numbers \( \mathbf{R} = \mathbf{Q} \cup \mathbf{Q}’ \), where \( \mathbf{Q}’ \) is the set of all irrational numbers.

Illustration 3

For example, let A be the set of numbers in the interval [3, 5] and B be the set of numbers in the interval [2,3]. Then the Cartesian product \( A \times B \) corresponds to the rectangular region shown in the Fig. 1.4. It consists of all points (x, y) within the region.

Fig. 1.4

Progress check

1. For any two non-empty sets A and B, \( A \times B \) is called as ______.
2. If \( n(A \times B) = 20 \) and \( n(A) = 5 \) then \( n(B) \) is ______.
3. If \( A = \{-1,1\} \) and \( B = \{-1,1\} \) then geometrically describe the set of points of \( A \times B \).
4. If A, B are the line segments given by the intervals (–4, 3) and (–2, 3) respectively, represent the cartesian product of A and B.

Note:

The set of all points in the cartesian plane can be viewed as the set of all ordered pairs (x, y) where x, y are real numbers. In fact, \( \mathbb{R} \times \mathbb{R} \) is the set of all points which we call as the cartesian plane.

Example 1.1

If \( A = \{1,3,5\} \) and \( B = \{2,3\} \) then (i) find \( A \times B \) and \( B \times A \).

(ii) Is \( A \times B = B \times A \)? If not why?

(iii) Show that \( n(A \times B) = n(B \times A) = n(A) \times n(B) \).

Solution

Given that \( A = \{1,3,5\} \) and \( B = \{2,3\} \)

(i) \( A \times B = \{1,3,5\} \times \{2,3\} = \{(1,2), (1,3), (3,2), (3,3), (5,2), (5,3)\} \) ...(1)

\( B \times A = \{2,3\} \times \{1,3,5\} = \{(2,1), (2,3), (2,5), (3,1), (3,3), (3,5)\} \) ...(2)

(ii) From (1) and (2) we conclude that \( A \times B \neq B \times A \) as \( (1, 2) \neq (2, 1) \) and \( (1, 3) \neq (3, 1) \), etc.

(iii) \( n(A)=3 \); \( n(B) = 2 \).

From (1) and (2) we observe that, \( n(A \times B) = n(B \times A) = 6 \);

we see that, \( n(A) \times n(B) = 3 \times 2 = 6 \) and \( n(B) \times n(A) = 2 \times 3 = 6 \)

Hence, \( n(A \times B) = n(B \times A) = n(A) \times n(B) = 6 \).

Thus, \( n(A \times B) = n(B \times A) = n(A) \times n(B) \).

Example 1.2

If \( A \times B = \{(3,2), (3,4), (5,2), (5,4)\} \) then find A and B.

Solution

\( A \times B = \{(3,2), (3,4), (5,2), (5,4)\} \)

We have A = {set of all first coordinates of elements of \( A \times B \)}. Therefore, A = {3,5}

B = {set of all second coordinates of elements of \( A \times B \)}. Therefore, B = {2,4}

Thus \( A = \{3,5\} \) and \( B = \{2,4\} \).

Example 1.3

Let \( A = \{x \in \mathbf{N} \mid 1 < x < 4\} \), \( B = \{x \in \mathbf{W} \mid 0 \leq x < 2\} \) and \( C = \{x \in \mathbf{N} \mid x < 3\} \).

Then verify that

(i) \( A \times (B \cup C) = (A \times B) \cup (A \times C) \)

(ii) \( A \times (B \cap C) = (A \times B) \cap (A \times C) \)

Solution

\( A = \{x \in \mathbf{N} \mid 1 < x < 4\} = \{2, 3\} \),

\( B = \{x \in \mathbf{W} \mid 0 \leq x < 2\} = \{0, 1\} \),

\( C = \{x \in \mathbf{N} \mid x < 3\} = \{1, 2\} \)

(i) \( A \times (B \cup C) = (A \times B) \cup (A \times C) \)

\( B \cup C = \{0, 1\} \cup \{1, 2\} = \{0, 1, 2\} \)

\( A \times (B \cup C) = \{2, 3\} \times \{0, 1, 2\} = \{(2, 0), (2, 1), (2, 2), (3, 0), (3, 1), (3, 2)\} \) ...(1)

\( A \times B = \{2, 3\} \times \{0, 1\} = \{(2,0),(2,1),(3,0),(3,1)\} \)

\( A \times C = \{2, 3\} \times \{1, 2\} = \{(2, 1), (2, 2), (3, 1), (3, 2)\} \)

\( (A \times B) \cup (A \times C) = \{(2, 0), (2, 1), (3, 0), (3, 1)\} \cup \{(2, 1), (2, 2), (3, 1), (3, 2)\} \)

\( = \{(2, 0), (2, 1), (2, 2), (3, 0), (3, 1), (3, 2)\} \) ...(2)

From (1) and (2), \( A \times (B \cup C) = (A \times B) \cup (A \times C) \) is verified.

(ii) \( A \times (B \cap C) = (A \times B) \cap (A \times C) \)

\( B \cap C = \{0, 1\} \cap \{1, 2\} = \{1\} \)

\( A \times (B \cap C) = \{2, 3\} \times \{1\} = \{(2,1),(3,1)\} \) ... (3)

\( A \times B = \{2, 3\} \times \{0, 1\} = \{(2, 0),(2, 1),(3, 0),(3, 1)\} \)

\( A \times C = \{2, 3\} \times \{1, 2\} = \{(2, 1),(2, 2),(3, 1),(3, 2)\} \)

\( (A \times B) \cap (A \times C) = \{(2, 0),(2, 1),(3, 0),(3, 1)\} \cap \{(2, 1),(2, 2),(3, 1),(3, 2)\} \)

\( = \{(2, 1),(3, 1)\} \) ... (4)

From (3) and (4), \( A \times (B \cap C) = (A \times B) \cap (A \times C) \) is verified.

Note

The above two verified properties are called distributive property of cartesian product over union and intersection respectively. In fact, for any three sets A, B, C we have

(i) \( A \times (B \cup C) = (A \times B) \cup (A \times C) \)

(ii) \( A \times (B \cap C) = (A \times B) \cap (A \times C) \)

Cartesian Product of three Sets

If A, B, C are three non-empty sets then the cartesian product of three sets is the set of all possible ordered triplets given by

\( A \times B \times C = \{(a,b,c) \text{ for all } a \in A, b \in B, c \in C \} \)

Illustration for Geometrical understanding of cartesian product of two and three sets

Let \( A = \{0,1\} \), \( B = \{0,1\} \), \( C = \{0,1\} \)

\( A \times B = \{0,1\} \times \{0,1\} = \{(0, 0),(0,1),(1, 0),(1,1)\} \)

Fig. 1.5

Representing \( A \times B \) in the xy - plane we get a picture shown in Fig. 1.5.

\( (A \times B) \times C = \{(0, 0),(0,1),(1, 0),(1,1)\} \times \{0,1\} \)

\( = \{(0, 0, 0),(0, 0,1),(0,1, 0),(0,1,1),(1, 0, 0),(1, 0,1)(1,1, 0),(1,1,1)\} \)

Representing \( A \times B \times C \) in the xyz - plane we get a picture as shown in Fig. 1.6

Fig. 1.6

Thus, \( A \times B \) represent vertices of a square in two dimensions and \( A \times B \times C \) represent vertices of a cube in three dimensions.

NOTES

In general, cartesian product of two non-empty sets provides a shape in two dimensions and cartesian product of three non-empty sets provide an object in three dimensions.

10th Maths Unit 1: Relations and Functions Exercise 1.1 Solutions | Cartesian Product

Exercise 1.1: Cartesian Product - Solutions

Maths Book back answers and solution for Exercise questions - Mathematics : Relation and Function: Cartesian Product: Exercise Questions with Answer, Solution

Question 1

Find \(A \times B\) , \(A \times A\) and \(B \times A\)

(i) \(A = \{2, -2, 3\}\) and \(B = \{1, -4\}\)

(ii) \(A = B = \{p, q\}\)

(iii) \(A = \{m, n\}\) ; \(B = \emptyset\)

Solution:

Question 2

Let \(A = \{1,2,3\}\) and \(B = \{x | x\) is a prime number less than 10}. Find \(A \times B\) and \(B \times A\).

Solution:

Question 3

If \(B \times A = \{(-2, 3), (-2, 4), (0, 3), (0, 4), (3, 3), (3, 4)\}\) find A and B.

Solution:

Question 4

If \(A = \{5, 6\}\) , \(B = \{4, 5, 6\}\) , \(C = \{5, 6, 7\}\) , Show that \(A \times A = (B \times B) \cap (C \times C)\) .

Solution:

Question 5

Given A={1,2,3}, \(B = \{2,3,5\}\), \(C = \{3,4\}\) and \(D = \{1,3,5\}\), check if \((A \cap C) \times (B \cap D) = (A \times B) \cap (C \times D)\) is true?

Solution:

Question 6

Let \(A = \{x \in W | x < 2\}\) , \(B = \{x \in N | 1 < x \le 4\}\) and \(C = \{3, 5\}\). Verify that

(i) \(A \times (B \cup C) = (A \times B) \cup (A \times C)\)

(ii) \(A \times (B \cap C) = (A \times B) \cap (A \times C)\)

(iii) \((A \cup B) \times C = (A \times C) \cup (B \times C)\)

Solution:

Question 7

Let A = The set of all natural numbers less than 8, B = The set of all prime numbers less than 8, C = The set of even prime number. Verify that

(i) \((A \cap B) \times C = (A \times C) \cap (B \times C)\)

(ii) \(A \times (B - C) = (A \times B) - (A \times C)\)

Solution:

Final Answers

• 1. (i) \(A \times B = \{(2, 1), (2, -4), (-2, 1), (-2, -4), (3, 1), (3, -4)\}\)
  \(A \times A = \{(2, 2), (2, -2), (2, 3), (-2, 2), (-2, -2), (-2, 3), (3, 2), (3, -2), (3, 3)\}\)
  \(B \times A = \{(1, 2), (1, -2), (1, 3), (-4, 2), (-4, -2), (-4, 3)\}\)
• 1. (ii) \(A \times B = \{(p, p), (p, q), (q, p), (q, q)\}\)
  \(A \times A = \{(p, p), (p, q), (q, p), (q, q)\}\)
  \(B \times A = \{(p, p), (p, q), (q, p), (q, q)\}\)
• 1. (iii) \(A \times B = \{\}\)
  \(A \times A = \{(m, m), (m, n), (n, m), (n, n)\}\)
  \(B \times A = \{\}\)
• 2. \(A \times B = \{(1, 2), (1, 3), (1, 5), (1, 7), (2, 2), (2, 3), (2, 5), (2, 7), (3, 2), (3, 3), (3, 5), (3, 7)\}\)
  \(B \times A = \{(2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3), (5, 1), (5, 2), (5, 3), (7, 1), (7, 2), (7, 3)\}\)
• 3. \(A = \{3, 4\}\) and \(B = \{-2, 0, 3\}\)
• 5. True

Introduction to Relations and Functions in Mathematics for Class 10

Introduction - Relation and Function | Mathematics

Relation and Function

Gottfried Wilhelm Leibniz (also known as von Leibniz) was a prominent German mathematician, philosopher, physicist and inventor. He wrote extensively on 26 topics covering wide range of subjects among which were Geology, Medicine, Biology, Epidemiology, Paleontology, Psychology, Engineering, Philology, Sociology, Ethics, History, Politics, Law and Music Theory

In a manuscript Leibniz used the word “function” to mean any quantity varying from point to point of a curve. Leibniz provided the foundations of Formal Logic and Boolean Algebra, which are fundamental for modern day computers. For all his remarkable discoveries and contributions in various fields, Leibniz is hailed as “The Father of Applied Sciences”.

Gottfried Wilhelm Leibniz

Learning Outcomes

• To define and determine cartesian product of sets.
• To define a relation as a subset of cartesian product of sets.
• To understand function as a special relation.
• To represent a function through an arrow diagram, a set of ordered pairs, a table, a rule or a graph.
• To classify functions as one-one, many-one, onto, into and bijection.
• To study combination of functions through composition operation.
• To understand the graphs of linear, quadratic, cubic and reciprocal functions.

Introduction

The notion of sets provides the stimulus for learning higher concepts in mathematics. A set is a collection of well-defined distinguishable objects. This means that a set is merely a collection of something which we may recognize. In this chapter, we try to extend the concept of sets in two forms called Relations and Functions. For doing this, we need to first know about cartesian products that can be defined between two non-empty sets.

It is quite interesting to note that most of the day-to-day situations can be represented mathematically either through a relation or a function. For example, the distance travelled by a vehicle in given time can be represented as a function. The price of a commodity can be expressed as a function in terms of its demand. The area of polygons and volume of common objects like circle, right circular cone, right circular cylinder, sphere can be expressed as a function with one or more variables.

In class IX, we had studied the concept of sets. We have also seen how to form new sets from the given sets by taking union, intersection and complementation.

Now we are about to study a new set called “cartesian product” for the given sets