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

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.