## Ordered pairs

Let A and B be two non-empty sets.

If a ∈ A , b ∈ B then (a, b) is an ordered pair.

'a' is called the first component and 'b' is called the second component of the ordered pair (a, b).

'a' and 'b' are also called as first coordinate and second coordinate respectively.

## Cartesian product

Let A and B be two non-empty sets.

Then {(a, b)/ a ∈ A ,b ∈ B} is called the Cartesian product of A and B, and is denoted by A × B. It is read as "A cross B" or as "A into B".

**Ex : ** Let A = {1, 2, 3}, B = {a, b}

A × B = {(1, a), (1, b), (2, a), (2, b), (3, a), (3, b)}

B × A = {(a, 1), (a, 2), (a, 3), (b, 1), (b, 2), (b, 3)}

Evidently the two are not the same.

But the number of ordered pairs in each is same i.e, 6.

**Note:**

(i) If A and B are two non-empty sets, then A × B ≠ B × A but n(A × B) = n(B × A).

(ii) If one of the sets A and B is empty, then A × B is also empty.

## Number system

(i) R is the set of all real numbers.

(ii) R^{+} is the set of all positive real numbers.

(iii) Q is the set of all rational numbers.

(iv) Q^{+} is the set of all positive rational numbers.

(v) N is the set of all natural numbers.

(vi) Z is the set of all integers.

(vii) Z^{+} is the set of all positive integers.

(viii) Z^{–} is the set of all negative integers.

(ix) W is the set of all whole numbers.

Let A and B be two non-empty sets.

If f ⊆ A × B, then f is called a relation from A into B (or simple A to B).

In particular, any relation from A to A is called a binary relation on A.

** Ex : ** Let A = {1, 2, 3} and B = {a, b}

We have

A × B = {(1, a), (1, b), (2, a), (2, b), (3, a), (3, b)}

B × A = {(a, 1), (a, 2), (a, 3), (b, 1), (b, 2), (b, 3)}

A × A = {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3,3)}

B × B = {(a, a), (a, b), (b, a), (b, b)}

(i) f = {(1, a)} is a relation from A to B, because f ⊆ A × B.

(ii) g = {(1, b), (2, a), (2, b), (3, b)} is also a relation from A to B, because of g ⊆ A × B.

(iii) h = {(a, 1), (b, 2)} is not a relation from A to B, but it is a relation from B to A.

(iv) i = {(1, 3), (2, 2), (3, 1), (3,3)} is a binary relation on A.

(v) j = {(a, b), (b, a)} is a binary relation on B.

**Definition of a function: **Let A and B be two non-empty sets. A function 'f' from A to B, written as f : A B, is a relation in which every element 'x' of A corresponds to only one element f(x) of B. The element f(x) is called the **image** of 'x' and the element 'x' is called the **pre-image** of f(x). The ordered pairs of a function 'f' are represented as (x, f(x)).

**Domain, co-domain and range of a function: **Let 'f' be a function from non-empty set A to another non-empty set B. Then, the non-empty set A is called the ‘domain’ of the function 'f', the non-empty set B is called the ‘co-domain’ of the function 'f' and {f(x): x A} B is called the ‘range’ of the function. In some cases, the range happens to be identical with the co–domain of a function.

**Ex:**Let A = { – 1, 1, 2, 3, 4} and B = {1, 4, 9, 16, 24} and the rule f(x) = x^{2}, x A is a function from A to B then find domain, co–domain and range of a function.

**Sol:**Given, f: A B is a function. Using the rule f(x) = x^{2}, x A we have:

f(–1) = (–1)^{2}= 1; f(1) = (1)^{2}= 1; f(2) = (2)^{2}= 4; f(3) = (3)^{2}= 9; f(4) = (4)^{2}= 16

∴ Domain of a function = A = {– 1, 1, 2, 3, 4};

Co-domain of a function = B = {1, 4, 9, 16, 24};

Range of a function = {1, 4, 9, 16}.

**Ex: **

1. |
2. |
3. |

This is not a function. Because the vertical line cuts the curve at two points | This is a function. Because the vertical line cuts the curve at only one point | This is also a function. Because the vertical line cuts the curve at only one point |

There are six ways to represent a function. They are: (i) roster form, (ii) set builder form, (iii) by formula, (iv) by table, (v) by arrow diagram and (vi) by graph.

**Roster form:**In this form, the function is represented by the set of all ordered pairs which belong to the function. For example, let A = {3, 6, 9}, B = {1, 2, 3} and f be the function "is thrice of" from A to B. Then, f = {(3, 1), (6, 2), (9, 3)}.**Set builder form:**In this form, the function is represented as: f = {(x, f(x)) : f(x) = x + 2}. For example, let f = {(3, 2), (6, 5), (9, 8)}. Then, set builder form of a function f is: f = {(x, f(x)) : f(x) = x – 1}.**By formula:**In this form, a formula, i.e., an algebraic equation can be used to represent a function. For example, an equation f(x) = x + 4 represents a function where x takes all values on set of natural numbers (N) and the values of f(x) are obtained by using the above function.**By table:**In this form, a table can be used to represent a function. For example, the table given below represents a function: f(x) = x + 4.**By arrow diagram:**In this form, the function is represented by drawing arrows from the first elements to the second elements of all ordered pairs which belong to the given function.**By graph:**In this form, the function is represented by drawing dots in the graph for all ordered pairs which satisfy the given function.

x | 1 | 2 | 3 | 4 | 5 |

f(x) | 5 | 6 | 7 | 8 | 9 |

## Many-one Function

A function f : A B is said to be many-one, if two or more than two elements in A have the same image in B, i.e., f : A B is a many-one function, if there exists x, y A such that x ≠ y but f(x) = f(y).

**Note : ** All even functions are many-one.

A function f : A B is said to be onto, if every element in B has at least one pre-image in A, i.e., f : A B is onto if and only if for every y B there exists at least one x A such that f(x) = y. An onto function is also called a surjective function or surjection.

**or**

A function f : A → B is said to be "onto" if the range of f is equal to the co-domain of f.

i.e, Range of f = f (A) = B (co-domain).

## Into Function

A function f : A → B is an into function if there exists an element
in B having no pre-image in A.

In other words, f : A → B is an into function if it is not an onto function.

## Method to check onto or into function:

Let f (x) be a function defined from A to B.

Step-1: Solve f(x) = y by taking ' x ' as a function of y say g(y).

Step-2: Now if g(y) is defined for each y ∈
B (co-domain) and g(y) ∈ A (domain), then f (x) is onto.

Step-3 : If any one of the two requirements is not fulfilled, then f(x) is into.

The **inverse** of a function is the set of ordered pairs obtained by interchanging the first and second elements of each ordered pair in the given function. If ‘f’ is a given function, then its inverse is denoted by **f ^{ -1}**.

The domain of ‘f’ is the range of f^{ -1} and the range of ‘f’ is the domain of f^{ -1}. An important point to remember is that: if 'f' is a function from A to B, then it is not necessary that f^{ –1} is a function from B to A.

Properties of inverse of a function |
---|

• The inverse of a bijective function is unique. |

• The inverse of a bijective function is bijective function. |

• If f : A B and g : B C are two bijective functions, then gof : A C is a bijective function and (gof)^{ -1} = f^{ -1}og^{ -1}. |

• If f : A → B is a bijection then fof^{ – 1 } = I_{B}, f^{ – 1}of = I_{A}. |

• If f : A → B and g : B → A are functions such that gof = I_{A} and fog = I_{B} then f : A → B is a bijection and g = f^{ – 1} |

• If f : A → B is a bijection fof ^{ – 1 } = f^{ – 1 }of = I_{A}. |

**Inverse of a function defined by an equation: **To find the inverse of a function defined by an equation, follow the procedure given below:

Step 1: Let f(x) = y. |

Step 2: Solve the equation y = f(x) for 'x'. The result is an equation of the form x = f^{ -1}(y). |

Step 3: Interchange 'x' and 'y' in the equation found in step 2. This expresses f^{ -1} as a function of 'x'. |

## 1. Identity function

A function f: A → B is said to be an Identity function if f(x) = x for all x ∈ A that means it always returns the same value that is used as its input. Identity function is denoted by I (first letter of Identity). Refer adjacent fig .(i)

## 2. Constant function

A function f: A → B is said to be a constant function if there is an element m ∈ B such that f(x) = m for all elements belongs to A. That means the range of the function contains only one element.

**Example: ** Let f : A → B is a function defined as f(x) = m as x ∈ A, that is, f(p) = m, f(q) = m, f(r) = m, f(s) = m, f(t) = m. So, 'f' is a constant function defined from A → B as f(x) = m for all x ∈ A. Refer adjacent fig .(ii)

## 3. Equal functions

Two functions f and g defined on the same domain D are said to be equal, if f(x) = g(x) for all x ∈ D. For example: let f and g are two functions defined as f(x) = x + 2 for x ∈ N and g(x) = x + 2 for x ∈ N, then f and g are called equal functions.