Relations and Functions

# Relations and Functions

## Ordered Pair

An ordered pair (x, y) is a pair of objects in which x is called the first component and y is called the second component. For example, if there are two numbers 4 and 5, then (4, 5) forms one ordered pair and (5, 4) forms another ordered pair.

## Properties of Ordered Pair

1. Order of the components is very important in an ordered pair. For any numbers a and b, (a, b) is not equal to (b, a). For example, (4, 7) is not equal to (7, 4).
2. Both the components of an ordered pair can be same. For any number a, (a, a) can be the ordered pair. For example, (7, 7), (9, 9), etc.
3. Two ordered pairs are equal if the first component of one ordered pair is equal to the first component of the second ordered pair. Same is true for the second component. For example, (11, 2) = (11, 2) but (11, 2) (11, 3).

## Cartesian Product

The cartesian product of two non-empty sets P and Q is the set of all ordered pairs whose first component is an element of set A and the second component is an element of set B. A cartesian product is shown as P × Q and is read as ‘P cross Q’.
For example, if P = {2, 4, 6} and Q = {5, 10, 15, 20} then P × Q = {(2, 5), (2, 10), (2, 15), (2, 20), (4, 5), (4, 10), (4, 15), (4, 20), (6, 5), (6, 10), (6, 15), (6, 20)}
Note that
1. P × Q Q × P when P Q. For example, if P = {2, 3}, Q = {5, 6}, then P × Q = {(2, 5), (2, 6), (3, 5), (3, 6)} but Q × P = {(5, 2), (5, 3), (6, 2), (6, 3)}.
2. Conversely, if P = Q, then P × Q = Q × P. For example, if P = {4, 5} and Q = {4, 5}, then P × Q = {(4, 4), (4, 5), (5, 4), (5, 5)} and Q × P = {(4, 4), (4, 5), (5, 4), (5, 5)}.
Thus, P × Q = Q × P.
3. Number of ordered pair is same in P × Q and Q × P, i.e., n (P × Q) = n (Q × P).              For example, if P = {8, 10} and Q = {9, 11}, then n (P × Q) = 4 and n (Q × P) = 4.            Thus, n (P × Q) = n (Q × P).

## Relation

We can see some kind of a connection or association between two objects in our daily life. For example,
a. Anu is elder than Sam. It shows the connection between two persons.
b. 1 is greater than –2. It shows the connection between two numbers.
Thus, the connection or association between two objects is defined as relation. Mathematically, any set of ordered pair is called a relation. Relation is generally represented using the letter R. Therefore in the first example, relation R = ‘is elder than’ and in the second example, relation R = ‘is greater than’.
Let A and B be two non-empty sets. Then the relation between an element a of set A and an element b of set B is written as a R b. For example, let P = {0, 2, 3, 4}, Q = {9, 16} and R be the relation defined from P to Q as ‘is a factor of’, then we get 3 R 9, 2 R 16, 4 R 16.

## Representation of a Relation

There are three methods of representing relation.
1. Roster form
2. Set builder form
3. Arrow diagram

### Roster Form

In this type of representation, a relation R from set A to set B is shown by the set of all ordered pairs (a, b) such that a Ïµ A and b Ïµ B which satisfies the given relation R.
Let A = {3, 7, 9, 15}, B = {2, 8} and R be the relation ‘is more than’ from A to B, then we get R = {(3, 2), (7, 2), (9, 2), (15, 2), (9, 8), (15, 8)}

### Set Builder Form or Equation Form

In this method, the relation R from A and B is represented using statements or formulae in which an element a Ïµ A and an element b Ïµ B.
Let A and B be two non-empty sets, then the relation R ‘is a square of’ from A to B is shown as R = {(a, b) : a Ïµ A, b Ïµ B and a is a square of b}.

### Arrow Diagram

In this method, the relation R from A to B is represented using arrows from set A to set B satisfying the given relation.

Let A and B be two non-empty sets, then the relation R ‘is a cube root of’ from A to B where A = {2, 3, 4, 5} and B = {8, 27, 64, 125} is represented as shown.

### Domain and Range of a Relation

The set of first components of the ordered pairs in a relation is its domain and the set of second components of the ordered pairs in a relation is its range.
Let X = {1, 2, 3, 4, 5, 6, 7}, Y = {1, 4, 9, 16, 18} and the relation R ‘is a square root of’ from X to Y, then R = {(1, 1), (2, 4), (3, 9), (4, 16)}, domain = {1, 2, 3, 4} and range = {1, 4, 9, 16}.

Example 1: Let P = {1, 2, 3, 4, 5, 6}, Q = {16, 24, 33} and the relation R ‘is a prime factor of’ from P to Q. Then
a. represent R in roster form, set builder form and using arrow diagram.
b. find the domain and range of R.

Solution: Given, P = {1, 2, 3, 4, 5, 6}, Q = {16, 24, 33} and relation R ‘is a prime factor of’ from P to Q.
Roster form: R = {(2, 16), (2, 24), (3, 24), (3, 33)}
Set builder form: R = {(p, q): p Ïµ P, q Ïµ Q and p is a prime factor of q}
Arrow diagram:

b. Domain of R = {2, 3}, Range of R = {16, 24, 33}

## Function or Mapping

A function or mapping from A to B is a special type of relation where A, B be two non-empty sets in which every element a of set A is related to a unique element b of set B. A function is written as f: A B, i.e., f is a function from A to B and it means that f is a function or mapping from A to B and is denoted by f (x). Also, set A is called the domain of the function and set B is the range of the function which is also known as co-domain.

## Conditions for Functions or Mapping

1. Every element of set A is associated with one element of set B, i.e., there should not be any element of set A which is not associated with any element of set B and no element of set A should be associated with 2 or more elements of set B.
2. If f is a function from A to B (f : A B), then no two ordered pairs should have the same first component.

## Representation of a Function

Let A and B be two non-empty sets and f : A B is represented as:

### Roster Form

Let A = {1, 8, 27, 64}, B = {1, 2, 3, 4, 5} and function f : A B have relation ‘is cube of’ is represented in roster form as f = {(1, 1), (8, 2), (27, 3), (64, 4)}.

### Equation Form

Let A = {1, 8, 27, 64}, B = {1, 2, 3, 4, 5} and function f : A B have relation ‘is cube of’ is represented in equation form as f (x) = y = 3x, where x Ïµ A and y Ïµ B.

### Set Builder Form

Let A = {1, 8, 27, 64}, B = {1, 2, 3, 4, 5} and function f : A B have relation ‘is cube of’ is represented in set builder form as f = {(x, y): x Ïµ A, y Ïµ B and y = 3x}.

### Arrow Diagram

Let A = {1, 8, 27, 64}, B = {1, 2, 3, 4, 5} and function f : A B have relation ‘is cube of’ is represented in arrow diagram as:

## Value of the Function

Let A and B be two non-empty sets, then f(x) is known as the value of the function at x. If f : x 3x – 4, then f(x) = 3x – 4
Thus, the value of the function at x = 3 is f(3) = 3 × 3 – 4 = 5.

Example 2: State whether the relations shown below are functions or not.
a. If A = {a, b, c, d} and B = {x, y, z}, then R = {(a, x), (b, y), (c, x), (d, y)}
b. If A = {1, 2, 3, 4, 5} and B = {6, 7}, then R = {(1, 6), (2, 7), (3, 7), (4, 7)}
c. If A = {10, 20, 30, 40, 50} and B = {a, b, c}, then R = {(10, b), (20, b), (30, b), (40, b), (50, c)}

Solution:
a. The relation R is a function as every element of set A is related to a unique element of set B.
b. The relation R is not a function as 5, an element of A is not related to any element of set B.
c. The relation R is a function as every element of set A is related to an element of set B.

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