Identifying Graphs of Linear, Quadratic, Cubic, and Reciprocal Functions

10th Mathematics : UNIT 1 : Relation and Function

Identifying the graphs of Linear, Quadratic, Cubic and Reciprocal functions

Study Material, Lecturing Notes, Assignment, Reference, Wiki description explanation, brief detail

Graphs provide visualization of curves and functions. Hence, graphs help a lot in understanding the concepts in a much efficient way.

In this section, we will be discussing about the identification of some of the functions through their graphs. In particular, we discuss graphs of Linear, Quadratic, Cubic and Reciprocal functions.

1. Linear Function

A function $f: \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x) = mx + c$, $m \neq 0$ is called a linear function. Geometrically this represents a straight line in the graph.

Some Specific Linear Functions and their graphs are given below.

2. Modulus or Absolute valued Function

A function $f: \mathbb{R} \rightarrow [0, \infty)$ defined by $f(x) = |x|$.

Note

• Modulus function is not a linear function but it is composed of two linear functions x and –x.
• Linear functions are always one-one functions and has applications in Cryptography as well as in several branches of Science and Technology.

3. Quadratic Function

A function $f: \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x) = ax^2 + bx + c$, $(a \neq 0)$ is called a quadratic function.

Some specific quadratic functions and their graphs:

The equations of motion of a particle travelling under the influence of gravity is a quadratic function of time. These functions are not one – one. (Why?)

4. Cubic Function

A function $f: \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x) = ax^3 + bx^2 + cx + d$, $(a \neq 0)$ is called a cubic function. The graph of $f(x) = x^3$ is shown in Fig.1.48.

5. Reciprocal Function

A function $f: \mathbb{R} - \{0\} \rightarrow \mathbb{R}$ defined by $f(x) = \frac{1}{x}$ is called a reciprocal function (Fig.1.49).

6. Constant Function

A function $f: \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x) = c$, for all $x \in \mathbb{R}$ is called a constant function (Fig.1.50).

Composition of Functions: Definition, Illustration, Examples, and Solutions

Composition of Functions - Definition, Illustration, Example, Solution

Introduction to Composition of Functions

When a car driver depresses the accelerator pedal, it controls the flow of fuel which in turn influences the speed of the car. Likewise, the composition of two functions is a kind of ‘chain reaction’, where the functions act upon one after another.

Fig. 1.40

We can explain this further with the concept that a function is a ‘process’. If f and g are two functions then the composition g(f (x)) is formed in two steps.

1. Feed an input (say x) to f;
2. Feed the output f(x) to g to get g(f (x)) and call it gf(x).

Fig. 1.41

Illustration

Consider the set A of all students, who appeared in class X of Board Examination. Each student appearing in the Board Examination is assigned a roll number. In order to have confidentiality, the Board arranges to deface the roll number of each student and assigns a code number to each roll number.

Let A be the set of all students appearing for the board exam. B ⊆ N be the set all roll numbers and C ⊆ N be the set of all code numbers. This gives rise to two functions f: A → B and g: B → C given by b = f (a) be the roll number assigned to student a, c = g(b) be the code number assigned to roll number b, where a ∈ A , b ∈ B and c ∈ C.

We can write c = g(b) = g(f (a)).

Thus, by the combination of these two functions, each student is eventually attached a code number. This idea leads to the following definition.

Definition

Let f : A → B and g : B → C be two functions (Fig.1.42). Then the composition of f and g denoted by g o f is defined as the function g o f (x ) = g( f (x )) for all x ∈ A.

Fig. 1.42

Example 1.20

Find f o g and g o f when f (x) = 2x + 1 and g(x) = x² – 2

Solution

f (x) = 2x + 1 , g(x) = x² – 2

f o g(x) = f (g(x)) = f (x² − 2) = 2(x² − 2) + 1 = 2x² – 3

g o f (x) = g(f (x)) = g(2x + 1) = (2x + 1)² − 2 = 4x² + 4x – 1

Thus f o g = 2x² − 3, g o f = 4x² + 4x − 1. From the above, we see that f o g ≠ g o f .

Note

Generally, f o g ≠ g o f for any two functions f and g. So, composition of functions is not commutative.

Example 1.21

Represent the function f(x) = √(2x² − 5x + 3) as a composition of two functions.

Solution

We set f₁(x) = 2x² − 5x + 3 and f₂(x) = √x

Then,

Example 1.22

If f (x) = 3x − 2 , g(x) = 2x + k and if f o g = g o f , then find the value of k.

Solution

f(x) = 3x − 2 , g(x) = 2x + k

f o g(x) = f (g(x)) = f (2x + k) = 3(2x + k) − 2 = 6x + 3k – 2

Thus, f o g(x) = 6x + 3k – 2.

g o f (x) = g(3x − 2) = 2(3x − 2) + k

Thus, g o f (x) = 6x − 4 + k.

Given that f o g = g o f

Therefore, 6x + 3k − 2 = 6x − 4 + k

6x − 6x + 3k − k = −4 + 2 ⇒ 2k = −2 ⇒ k = −1

Example 1.23

Find k if f o f (k) = 5 where f (k) = 2k – 1.

Solution

f o f (k) = f (f (k))

= 2(2k − 1) − 1 = 4k − 3

Thus, f o f (k) = 4k – 3

But, it is given that f o f (k) = 5

Therefore 4k - 3 = 5 ⇒ 4k = 8 ⇒ k = 2.

Composition of three functions

Let A, B, C, D be four sets and let f : A → B , g : B → C and h : C → D be three functions (Fig.1.43). Using composite functions f o g and g o h, we get two new functions like (f o g) o h and f o (g o h).

Fig. 1.43

We observed that the composition of functions is not commutative. The natural question is about the associativity of the operation.

Note

Composition of three functions is always associative. That is, f o (g o h) = (f o g) o h

Example 1.24

If f(x) = 2x + 3, g(x) = 1 − 2x and h(x) = 3x. Prove that f o (g o h) = (f o g) o h

Solution

f(x) = 2x + 3 , g(x) = 1 − 2x , h(x) = 3x

Now, (f o g)(x) = f (g(x)) = f (1 − 2x) = 2(1 − 2x) + 3 = 5 − 4x

Then, (f o g) o h(x) = (f o g)(h(x)) = (f o g)(3x) = 5 − 4(3x) = 5 − 12x ……… (1)

(g o h)(x) = g(h(x)) = g(3x) = 1 − 2(3x) = 1 − 6x

So, f o (g o h)(x) = f (1 − 6x) = 2(1 − 6x ) + 3 = 5 − 12x ……… (2)

From (1) and (2), we get (f o g) o h = f o (g o h)

Example 1.25

Find x if gff(x) = fgg(x), given f (x) = 3x + 1 and g(x) = x + 3.

Solution

gff(x) = g [f {f (x)}] (This means “g of f of f of x”)

= g [ f (3x +1)] = g [ 3(3x +1)+1] = g (9x + 4)

g (9x + 4) = [ (9x + 4) + 3] = 9x + 7

fgg(x) = f [g {g (x)}] (This means “f of g of g of x”)

= f [ g (x + 3)] = f [ (x + 3) + 3] = f (x + 6)

f (x + 6) = [ 3(x + 6) + 1 ] = 3x + 19

These two quantities being equal, we get 9x + 7 = 3x + 19. Solving this equation we obtain x = 2.

Representation of Functions: A Guide for 10th Grade Mathematics

Representation of Functions - Mathematics

Representation of Functions

A function may be represented by

• (a) a set of ordered pairs
• (b) a table form
• (c) an arrow diagram
• (d) a graphical form

Let \(f: A \rightarrow B\) be a function

(a) Set of ordered pairs

The set \(f = \{(x, y) | y = f(x), x \in A\}\) of all ordered pairs represent a function.

(b) Table form

The values of \(x\) and the values of their respective images under \(f\) can be given in the form of a table.

(c) Arrow diagram

An arrow diagram indicates the elements of the domain of \(f\) and their respective images by means of arrows.

(d) Graph

The ordered pairs in the collection \(f = \{(x, y) | y = f(x), x \in A\}\) are plotted as points in the \(xy\)-plane. The graph of \(f\) is the totality of all such points.

Every function can be represented by a curve in a graph. But not every curve drawn in a graph will represent a function.

The following test will help us in determining whether a given curve is a function or not.

Vertical line test

“A curve drawn in a graph represents a function, if every vertical line intersects the curve in at most one point.”

Example 1.10

Using vertical line test, determine which of the following curves (Fig.1.18(a), 1.18(b), 1.18(c), 1.18(d)) represent a function?

Solution

The curves in Fig.1.18(a) and Fig.1.18(c) do not represent a function as the vertical lines meet the curves in two points \(P\) and \(Q\).

The curves in Fig.1.18(b) and Fig.1.18(d) represent a function as the vertical lines meet the curve in at most one point.

Notes

Any equation represented in a graph is usually called a ‘curve’.

Example 1.11

Let \(A = \{1, 2, 3, 4\}\) and \(B = \{2, 5, 8,11,14\}\) be two sets. Let \(f: A \rightarrow B\) be a function given by \(f(x) = 3x - 1\). Represent this function

1. by arrow diagram
2. in a table form
3. as a set of ordered pairs
4. in a graphical form

Solution

Given: \(A = \{1, 2, 3, 4\}\); \(B = \{2, 5, 8,11,14\}\); and \(f(x) = 3x - 1\).

\(f(1) = 3(1) - 1 = 3 - 1 = 2\)

\(f(2) = 3(2) - 1 = 6 - 1 = 5\)

\(f(3) = 3(3) - 1 = 9 - 1 = 8\)

\(f(4) = 3(4) - 1 = 12 - 1 = 11\)

(i) Arrow diagram

Let us represent the function \(f: A \rightarrow B\) by an arrow diagram (Fig.1.19).

Fig.1.19

(ii) Table form

The given function \(f\) can be represented in a tabular form as given below.

(iii) Set of ordered pairs

The function \(f\) can be represented as a set of ordered pairs as
\(f = \{(1,2),(2,5),(3,8),(4,11)\}\)

(iv) Graphical form

In the adjacent \(xy\)-plane the points (1,2), (2,5), (3,8), (4,11) are plotted (Fig.1.20).

Fig.1.20

Understanding Functions in Mathematics: Definitions, Illustrations, and Examples

Functions - Definition, Illustration, Example, Solution

Introduction to Functions

Among several relations that exist between two non-empty sets, some special relations are important for further exploration. Such relations are called “Functions”.

Illustration 8

A company has 5 employees in different categories. If we consider their salary distribution for a month as shown by arrow diagram in Fig.1.11, we see that there is only one salary associated for every employee of the company.

Here are various real life situations illustrating some special relations:

1. Consider the set A of all of your classmates; corresponding to each student, there is only one age.
2. You go to a shop to buy a book. If you take out a book, there is only one price corresponding to it; it does not have two prices corresponding to it. (of course, many books may have the same price).
3. You are aware of Boyle’s law. Corresponding to a given value of pressure P, there is only one value of volume V.
4. In Economics, the quantity demanded can be expressed as \(Q = 360 - 4P\), where P is the price of the commodity. We see that for each value of P, there is only one value of Q. Thus the quantity demanded Q depend on the price P of the commodity.

We often come across certain relations, in which, for a given element of a set A, there is only one corresponding element of a set B. Such relations are called functions. We usually use the symbol f to denote a functional relation.

Definition

A relation f between two non-empty sets X and Y is called a function from X to Y if, for each \(x \in X\) there exists only one \(y \in Y\) such that \((x, y) \in f\).

That is, \(f = \{(x,y)| \text{ for all } x \in X, y \in Y \}\).

A function f from X to Y is written as \(f: X \to Y\).

Comparing the definitions of relation and function, we see that every function is a relation. Thus, functions are subsets of relations and relations are subsets of cartesian product.

A function f can be thought as a mechanism (or device), which gives a unique output \(f(x)\) to every input x.

Notes

If \(f: X \to Y\) is a function then

• The set X is called the domain of the function f and the set Y is called its co-domain.
• If \(f(a) = b\), then b is called ‘image’ of a under f and a is called a ‘pre-image’ of b.
• The set of all images of the elements of X under f is called the ‘range’ of f.
• \(f: X \to Y\) is a function only if
  1. every element in the domain of f has an image.
  2. the image is unique.
• If A and B are finite sets such that \(n(A) = p\), \(n(B) = q\) then the total number of functions that exist between A and B is \(q^p\).
• In this chapter we always consider f to be a real valued function.
• Describing domain of a function
  1. Let \(f(x) = \frac{1}{x + 1}\). If \(x = -1\) then \(f(-1)\) is not defined. Hence f is defined for all real numbers except at \(x = -1\). So domain of f is \(\mathbf{R} - \{-1\}\).
  2. Let \(f(x) = \frac{1}{x^2 - 5x + 6}\). If \(x = 2, 3\) then \(f(2)\) and \(f(3)\) are not defined. Hence f is defined for all real numbers except at \(x = 2\) and \(3\). So domain of f = \(\mathbf{R} - \{2, 3\}\).

Illustration 9 - Testing for functions

Representation by Arrow diagram

Functions play very important role in the understanding of higher ideas in mathematics. They are basic tools to convert from one form to another form. In this sense, functions are widely applied in Engineering Sciences.

Note: The range of a function is a subset of it’s co-domain.

Example 1.6

Let \(X = \{1, 2, 3, 4\}\) and \(Y = \{2, 4, 6, 8,10\}\) and \(R = \{(1,2),(2,4),(3,6),(4,8)\}\). Show that R is a function and find its domain, co-domain and range?

Solution

Pictorial representation of R is given in Fig.1.14. From the diagram, we see that for each \(x \in X\), there exists only one \(y \in Y\). Thus all elements in X have only one image in Y. Therefore R is a function.

Domain X = {1,2,3,4}; Co-domain Y = {2,3,6,8,10}; Range of f = {2,4,6,8}.

Example 1.7

A relation ‘f’ is defined by \(f(x) = x^2 - 2\) where, \(x \in \{-2, -1, 0, 3\}\)

(i) List the elements of f

(ii) Is f a function?

Solution

\(f(x) = x^2 - 2\) where \(x \in \{-2, -1, 0, 3\}\)

(i) \(f(-2) = (-2)^2 - 2 = 2\); \(f(-1) = (-1)^2 - 2 = -1\)

\(f(0) = (0)^2 - 2 = -2\); \(f(3) = (3)^2 - 2 = 7\)

Therefore, f = {(-2, 2), (-1,-1), (0,-2), (3, 7)}

(ii) We note that each element in the domain of f has a unique image. Therefore f is a function.

Example 1.8

If \(X = \{-5,1,3,4\}\) and \(Y = \{a,b,c\}\), then which of the following relations are functions from X to Y?

(i) \(R_1 = \{(-5,a), (1,a), (3,b)\}\)

(ii) \(R_2 = \{(-5,b), (1,b), (3,a),(4,c)\}\)

(iii) \(R_3 = \{(-5,a), (1,a), (3,b),(4,c),(1,b)\}\)

Solution

(i) \(R_1 = \{(-5,a), (1,a), (3,b)\}\)

We may represent the relation \(R_1\) in an arrow diagram (Fig.1.15(a)).

\(R_1\) is not a function as \(4 \in X\) does not have an image in Y.

(ii) \(R_2 = \{(-5,b), (1,b), (3,a),(4,c)\}\)

Arrow diagram of R2 is shown in Fig.1.15(b).

\(R_2\) is a function as each element of X has an unique image in Y.

(iii) \(R_3 = \{(-5,a), (1,a), (3,b),(4,c),(1,b)\}\)

Representing \(R_3\) in an arrow diagram (Fig.1.15(c)).

\(R_3\) is not a function as \(1 \in X\) has two images \(a \in Y\) and \(b \in Y\).

Note that the image of an element should always be unique.

Example 1.9

Given \(f(x) = 2x - x^2\),

find (i) \(f(1)\) (ii) \(f(x+1)\) (iii) \(f(x) + f(1)\)

Solution

(i) Replacing x with 1, we get

\(f(1) = 2(1) - (1)^2 = 2 - 1 = 1\)

(ii) Replacing x with \(x+1\), we get

\(f(x+1) = 2(x + 1) - (x + 1)^2 = 2x + 2 - (x^2 + 2x + 1) = -x^2 + 1\)

(iii) \(f(x) + f(1) = (2x - x^2) + 1 = -x^2 + 2x + 1\)

[Note that \(f(x) + f(1) \neq f(x + 1)\). In general \(f(a + b)\) is not equal to \(f(a) + f(b)\)]

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.

Ordered Pair: An Introduction to Relations and Functions | 10th Mathematics

Ordered Pair - Relation and Function | Mathematics

Ordered Pair

Observe the seating plan in an auditorium (Fig.1.1). To help orderly occupation of seats, tokens with numbers such as (1,5), (7,16), (3,4), (10,12) etc. are issued. The person who gets (4,10) will go to row 4 and occupy the 10^th seat. Thus the first number denotes the row and the second number, the seat. Which seat will the visitor with token (5,9) occupy? Can he go to 9^th row and take the 5^th seat? Do (9,5) and (5,9) refer to the same location? No, certainly! What can you say about the tokens (2,3), (6,3) and (10,3)?

This is one example where a pair of numbers, written in a particular order, precisely indicates a location. Such a number pair is called an ordered pair of numbers. This notion is skillfully used to mathematize the concept of a “Relation”.