12 class Maths Notes Chapter 1 Relations and Functions free PDF| Quick revision Notes class 12 maths
CBSE Revision Notes for CBSE Class 12 Mathematics Relations and Functions Types of relations: reflexive, symmetric, transitive and equivalence relations. One to one and onto functions, composite functions, inverse of a function. Binary operations.
Class 12 Maths Chapter-1 Relations and Functions Quick Revision Notes Free Pdf
✳️ Definitions
🔹Let A and B be two non-empty sets, then a function f from set A to set B is a rule which associates each element of A to a unique element of B.
✳️ Relation
🔹 If (a, b) € R, we say that a is related to b under the relation R and we write as a R b
✳️ Function
🔹 It is represented as f: A → B and function is also called mapping .
✳️ Real Function
🔹 f: A → B is called a real function, if A and B are subsets of R.
✳️ Domain and Codomain of a Real Function
🔹 Domain and codomain of a function f is a set of all real numbers x for which f(x) is a real number. Here, set A is domain and set B is codomain .
✳️ Range of a real function
🔹 f is a set of values f(x) which it attains on the points of its domain .
✳️ Types of Relations :-
🔹 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 x A.
🔹 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 x A.
🔹 Both the empty relation and the universal relation are sometimes called Trivial Relations .
🔹 A relation R in a set A is called .
👉 Reflexive
🔹 if (a, a) ∈ R, for every a ∈ A,
👉 Symmetric
🔹 If (a1, a2) ∈ R implies that (a2, a1) ∈ R, for all a1, a2∈ A.
👉 Transitive
🔹 If (a1, a2) ∈ R and (a2, a3) ∈ R implies that (a1, a3) ∈ R, for all a1, a2, a3 ∈ A.
🔹 A relation R in a set A is said to be an equivalence relation if R is reflexive, symmetric and transitive .
🔹 The set E of all even integers and the set O of all odd integers are subsets of Z satisfying following conditions:
👉 All elements of E are related to each other and all elements of O are related to each other.
👉 No element of E is related to any element of O and vice-versa.
👉 E and O are disjoint and Z = E U O.
👉 The subset E is called the equivalence class containing zero, Denoted by [0].
Ois the equivalence class containing 1 and is denoted by [1].
🔷 Note :-
🔹 [o] ≠ [1]
🔹 [o] = [2r]
🔹 [1] = [2r + 1], r ∈ Z.
🔷 Given an arbitrary equivalence relation R in an arbitrary set X, R divides X into mutually disjoint subsets Ai called partitions or subdivisions of X satisfying:
🔹 All elements of Ai are related to each other, for all i.
🔹 No element of Ai is related to any element of Aj, i ≠ j.
🔹 UAj = X and Ai∩Aj = φ, i + j.
🔷 The subsets Ai are called equivalence classes.
✳️ Note:
🔷 Two ways of representing a relation
🔹 Roaster method
🔹 Set builder method
🔷 If (a, b) ∈ R, we say that a is related to b and we denote it as a R b.
✳️ Types of Funetions
🔹 Consider the functions f1, f2, f3 and f4 given
🔹 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 .
🔹 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.
🔹 f: X→Y is onto if and only if Range of f = Y.
🔹 Eg :
🔹 A function f: X→Y is said to be one-one and onto (or bijective), if f is both one-one and onto.
🔹 Eg:
✳️ Composition of Functions and Invertible Function
✳️ Composite Function
🔹 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.
🔹 Eg:
● Let f : {2, 3, 4, 5} → {3, 4, 5, 9} and g : {3, 4, 5, 9} → {7, 11, 15} be functions
● Defined as f (2) = 3, f(3) = 4, f (4) = f (5) = 5 and g (3) = g (4) = 7 and g (5) = g (9) = 11.
● Find gof.
🔹 Solution
● gof(2) = g (f(2)) = g (3) = 7,
● gof (3) = g (f(3)) = g (4) = 7,
● gof (4) = g (f(4)) = g (5) = 11 and
● gof (5) = g (5) = 11
🔹 It can be verified in general that gof is one-one implies that f is one-one. Similarly, gof is onto implies that g is onto.
🔹 While composing f and g, to get gof, first f and then g was applied, while in the reverse process of the composite gof, first the reverse process of g is applied and then the reverse process of f.
🔹 If f: X → Y is a function such that there exists a function g: Y → X such that gof = IX and fog = IY, then f must be one-one and onto.
✳️ Invertible Function
🔹 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
🔹 Denoted by f ⁻¹.
🔹 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.
✳️ Theorem 1
🔹 If f : X→Y, g : Y→Z and h : Z→S are functions, then
• h₀ (g₀f) = (h₀g) ₀f.
🔹 Proof
We have
• h₀ (g₀f) (x) = h(g₀f (x)) = h(g(f(x))), ∀ x in X
• (h₀g)₀f (x) = h₀g (f(x)) = h (g (f(x))), ∀ x in X.
Hence, h₀ (g₀f) = (h₀g) ₀ f
✳️ Theorem 2
🔹Let f: X→Y and g: Y→ Z be two invertible functions.
• Then g₀f is also invertible with (g₀f)⁻¹ =f⁻¹₀ g⁻¹
🔹 Proof
• To show that g₀f is invertible with (g ₀ f)⁻¹ =f⁻¹₀g⁻¹, it is enough to show that (f⁻¹₀g⁻¹) o (g ₀ f) = Iâ‚“ and (g ₀ f) ₀ (f⁻¹ ₀ g⁻¹) = Iz.
Now, (f⁻¹₀g⁻¹) o (g ₀ f) = ((F⁻¹ ₀ g⁻¹) ₀ g) of, by Theorem 1
= (f⁻¹₀ (g⁻¹₀g)) of, by Theorem 1
= (f⁻¹ ₀ Iy) of, by definition of g⁻¹
= Iâ‚“
Similarly, it can be shown that (g ₀ f) ₀ (f⁻¹ ₀ g⁻¹) = Iz
✳️ Binary Operations
✳️ Definitions:
🔹 A binary operation * on a set A is a function * : A × A → A. We denote * (a, b) by a * b.
🔹A binary operation * on the set X is called commutative, if a * b = b * a, for every a, b ∈ X
🔹 A binary operation * : A × A→A is said to be associative if (a * b) * c = a * (b * c), ∀ a, b, c, ∈ A.
🔹 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.
• Zero is identity for the addition operation on R but it is not identity for the addition operation on N, as o ∈ N.
• Addition operation on N does not have any identity.
• For the addition operation + : R × R→R, given any a ∈ R, there exists - a in R such that a + (-a) = 0 (identity for '+') = (-a) + a.
🔹 A binary operation * : A × A + A with the identity element e in A, an element a e 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-
CBSE Revision Notes for CBSE Class 12 Mathematics Relations and Functions Types of relations: reflexive, symmetric, transitive and equivalence relations. One to one and onto functions, composite functions, inverse of a function. Binary operations.
Class 12 Maths Chapter-1 Relations and Functions Quick Revision Notes Free Pdf
✳️ Definitions
🔹Let A and B be two non-empty sets, then a function f from set A to set B is a rule which associates each element of A to a unique element of B.
✳️ Relation
🔹 If (a, b) € R, we say that a is related to b under the relation R and we write as a R b
✳️ Function
🔹 It is represented as f: A → B and function is also called mapping .
✳️ Real Function
🔹 f: A → B is called a real function, if A and B are subsets of R.
✳️ Domain and Codomain of a Real Function
🔹 Domain and codomain of a function f is a set of all real numbers x for which f(x) is a real number. Here, set A is domain and set B is codomain .
✳️ Range of a real function
🔹 f is a set of values f(x) which it attains on the points of its domain .
✳️ Types of Relations :-
🔹 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 x A.
🔹 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 x A.
🔹 Both the empty relation and the universal relation are sometimes called Trivial Relations .
🔹 A relation R in a set A is called .
👉 Reflexive
🔹 if (a, a) ∈ R, for every a ∈ A,
👉 Symmetric
🔹 If (a1, a2) ∈ R implies that (a2, a1) ∈ R, for all a1, a2∈ A.
👉 Transitive
🔹 If (a1, a2) ∈ R and (a2, a3) ∈ R implies that (a1, a3) ∈ R, for all a1, a2, a3 ∈ A.
🔹 A relation R in a set A is said to be an equivalence relation if R is reflexive, symmetric and transitive .
🔹 The set E of all even integers and the set O of all odd integers are subsets of Z satisfying following conditions:
👉 All elements of E are related to each other and all elements of O are related to each other.
👉 No element of E is related to any element of O and vice-versa.
👉 E and O are disjoint and Z = E U O.
👉 The subset E is called the equivalence class containing zero, Denoted by [0].
Ois the equivalence class containing 1 and is denoted by [1].
🔷 Note :-
🔹 [o] ≠ [1]
🔹 [o] = [2r]
🔹 [1] = [2r + 1], r ∈ Z.
🔷 Given an arbitrary equivalence relation R in an arbitrary set X, R divides X into mutually disjoint subsets Ai called partitions or subdivisions of X satisfying:
🔹 All elements of Ai are related to each other, for all i.
🔹 No element of Ai is related to any element of Aj, i ≠ j.
🔹 UAj = X and Ai∩Aj = φ, i + j.
🔷 The subsets Ai are called equivalence classes.
✳️ Note:
🔷 Two ways of representing a relation
🔹 Roaster method
🔹 Set builder method
🔷 If (a, b) ∈ R, we say that a is related to b and we denote it as a R b.
✳️ Types of Funetions
🔹 Consider the functions f1, f2, f3 and f4 given
🔹 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 .
🔹 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.
🔹 f: X→Y is onto if and only if Range of f = Y.
🔹 Eg :
🔹 A function f: X→Y is said to be one-one and onto (or bijective), if f is both one-one and onto.
🔹 Eg:
✳️ Composite Function
🔹 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.
🔹 Eg:
● Let f : {2, 3, 4, 5} → {3, 4, 5, 9} and g : {3, 4, 5, 9} → {7, 11, 15} be functions
● Defined as f (2) = 3, f(3) = 4, f (4) = f (5) = 5 and g (3) = g (4) = 7 and g (5) = g (9) = 11.
● Find gof.
🔹 Solution
● gof(2) = g (f(2)) = g (3) = 7,
● gof (3) = g (f(3)) = g (4) = 7,
● gof (4) = g (f(4)) = g (5) = 11 and
● gof (5) = g (5) = 11
🔹 It can be verified in general that gof is one-one implies that f is one-one. Similarly, gof is onto implies that g is onto.
🔹 While composing f and g, to get gof, first f and then g was applied, while in the reverse process of the composite gof, first the reverse process of g is applied and then the reverse process of f.
🔹 If f: X → Y is a function such that there exists a function g: Y → X such that gof = IX and fog = IY, then f must be one-one and onto.
✳️ Invertible Function
🔹 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
🔹 Denoted by f ⁻¹.
🔹 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.
✳️ Theorem 1
🔹 If f : X→Y, g : Y→Z and h : Z→S are functions, then
• h₀ (g₀f) = (h₀g) ₀f.
🔹 Proof
We have
• h₀ (g₀f) (x) = h(g₀f (x)) = h(g(f(x))), ∀ x in X
• (h₀g)₀f (x) = h₀g (f(x)) = h (g (f(x))), ∀ x in X.
Hence, h₀ (g₀f) = (h₀g) ₀ f
✳️ Theorem 2
🔹Let f: X→Y and g: Y→ Z be two invertible functions.
• Then g₀f is also invertible with (g₀f)⁻¹ =f⁻¹₀ g⁻¹
🔹 Proof
• To show that g₀f is invertible with (g ₀ f)⁻¹ =f⁻¹₀g⁻¹, it is enough to show that (f⁻¹₀g⁻¹) o (g ₀ f) = Iâ‚“ and (g ₀ f) ₀ (f⁻¹ ₀ g⁻¹) = Iz.
Now, (f⁻¹₀g⁻¹) o (g ₀ f) = ((F⁻¹ ₀ g⁻¹) ₀ g) of, by Theorem 1
= (f⁻¹₀ (g⁻¹₀g)) of, by Theorem 1
= (f⁻¹ ₀ Iy) of, by definition of g⁻¹
= Iâ‚“
Similarly, it can be shown that (g ₀ f) ₀ (f⁻¹ ₀ g⁻¹) = Iz
✳️ Binary Operations
✳️ Definitions:
🔹 A binary operation * on a set A is a function * : A × A → A. We denote * (a, b) by a * b.
🔹A binary operation * on the set X is called commutative, if a * b = b * a, for every a, b ∈ X
🔹 A binary operation * : A × A→A is said to be associative if (a * b) * c = a * (b * c), ∀ a, b, c, ∈ A.
🔹 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.
• Zero is identity for the addition operation on R but it is not identity for the addition operation on N, as o ∈ N.
• Addition operation on N does not have any identity.
• For the addition operation + : R × R→R, given any a ∈ R, there exists - a in R such that a + (-a) = 0 (identity for '+') = (-a) + a.
🔹 A binary operation * : A × A + A with the identity element e in A, an element a e 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-
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