CBSE Class 12 Maths Chapter 1 Relations and Functions Summary
In this chapter, there are following Relations and Functions Important points, which is first chapter of CBSE Class 12 Maths.
Relations and Function 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 Function Important Point 2:
A relation in a set A is a subset of A × A.
Relations and Function Important Point 3:
The empty set φ and A × A are two extreme relations.
Relations and Function 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 Function 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 Function 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.
