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# CBSE Class 12 Maths Chapter 1 Relations and Functions Summary

## Relations and Functions Important Point 1:

If (a, b) ∈ R, we say that a is related to b under the relation R and we write as a R b. In general, (a, b) ∈ R, we do not bother whether there is a recognizable connection or link between a and b.

## Relations and Functions Important Point 2:

A relation in a set A is a subset of A × A.

## Relations and Functions Important Point 3:

The empty set φ and A × A are two extreme relations.

## Relations and Functions Important Point 4:

A relation R in a set A is called empty relation, if no element of A is related to any element of A, i.e., R = φ ⊂ A × A.

## Relations and Functions Important Point 5:

A relation R in a set A is called universal relation, if each element of A is related to every element of A, i.e., R = A × A.

## Relations and Functions Important Point 6:

Both the empty relation and the universal relation are some times called trivial relations.

## Relations and Functions Important Point 7:

If (a, b) ∈ R, we say that a is related to b and we denote it as a R b.

## Relations and Functions Important Point 8:

A relation R in a set A is called (i) reflexive, if (a, a) ∈ R, for every a∈ A, (ii) symmetric, if (a1, a2) ∈ R implies that (a2, a1)∈ R, for all a1, a2 ∈ A. (iii) transitive, if (a1, a2) ∈ R and (a2, a3)∈ R implies that (a1, a3)∈ R, for all a1, a2, a3 ∈ A.

## Relations and Functions Important Point 9:

A relation R in a set A is said to be an equivalence relation if R is reflexive, symmetric and transitive.

## Relations and Functions Important Point 10:

A function f : X → Y is defined to be one-one (or injective), if the images of distinct elements of X under f are distinct, i.e., for every x1, x2 ∈ X, f (x1) = f (x2) implies x1 = x2. Otherwise, f is called many-one.

## Relations and Functions Important Point 11:

A function f : X → Y is said to be onto (or surjective), if every element of Y is the image of some element of X under f, i.e., for every y ∈ Y, there exists an element x in X such that f (x) = y.

## Relations and Functions Important Point 12:

The function f : X → Y is onto if and only if Range of f = Y.

## Relations and Functions Important Point 13:

A function f : X → Y is said to be one-one and onto (or bijective), if f is both one-one and onto.

## Relations and Functions Important Point 14:

In the following images, the function f1 in (i) and f4 in (iv) are one-one and the function f2 in (ii) and f3 in (iii) are many-one. The function f3 in (iii) and the function f4 in (iv) are onto and the function f1 in (i) is not onto as elements e, f in X2 are not the image of any element in X1 under f1. The function f4 in (iv) is one-one and onto.

## Relations and Functions Important Point 15:

Let f : A → B and g : B → C be two functions. Then the composition of f and g, denoted by gof, is defined as the function gof : A → C given by gof (x) = g(f (x)), ∀ x ∈ A.

## Relations and Functions Important Point 16:

A function f : X → Y is defined to be invertible, if there exists a function g : Y → X such that gof = IX and fog = IY. The function g is called the inverse of f and is denoted by f –1.

## Relations and Functions Important Point 17:

Thus, if f is invertible, then f must be one-one and onto and conversely, if f is one-one and onto, then f must be invertible.

## Relations and Functions Important Point 18:

If f : X → Y, g : Y → Z and h : Z → S are functions, then ho(gof ) = (hog)of.

## Relations and Functions Important Point 19:

Let f : X → Y and g : Y → Z be two invertible functions. Then gof is also invertible with (gof)–1 = f –1og–1.

## Relations and Functions Important Point 20:

A binary operation ∗ on a set A is a function ∗ : A × A → A. We denote ∗ (a, b) by a ∗ b.

## Relations and Functions Important Point 21:

A binary operation ∗ on the set X is called commutative, if a ∗ b = b ∗ a, for every a, b ∈ X.

## Relations and Functions Important Point 22:

A binary operation ∗ : A × A → A is said to be associative if (a ∗ b) ∗ c = a ∗ (b ∗ c), ∀ a, b, c, ∈ A.

## Relations and Functions Important Point 23:

Given a binary operation ∗ : A × A → A, an element e ∈ A, if it exists, is called identity for the operation ∗, if a ∗ e = a = e ∗ a, ∀ a ∈ A.

## Relations and Functions Important Point 24:

Zero is identity for the addition operation on R but it is not identity for the addition operation on N, as 0 ∉ N. In fact the addition operation on N does not have any identity.

## Relations and Functions Important Point 25:

Given a binary operation ∗ : A × A → A with the identity element e in A, an element a ∈ A is said to be invertible with respect to the operation ∗, if there exists an element b in A such that a ∗ b = e = b ∗ a and b is called the inverse of a and is denoted by a–1.

## Relations and Functions Important Point 26:

If R1 and R2 are equivalence relations in a set A, show that R1 ∩ R2 is also an equivalence relation.