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Relations and Functions For Class 12 Concepts
The topics and subtopics covered in relations and Functions for class 12 are:
Introduction
Types of Relations
Types of Functions
Composition of functions and invertible functions
Binary operations
Let us discuss the concept of relation and function in detail here.
Relation
The concept of relation is used in relating two objects or quantities with each other. Suppose two sets are considered, the relationship between them will be established if there is a connection between the elements of two or more non-empty sets.
Mathematically, “a relation R from a set A to a set B is a subset of the cartesian product A × B obtained by describing a relationship between the first element x and the second element y of the ordered pairs in A × B”.
Types of Relations
A relation R from A to A is also stated as a relation on A, and it can be said that the relation in a set A is a subset of A × A. Thus, the empty set φ and A × A are two extreme relations. Below are the definitions of types of relations:
Empty Relation
If no element of A is related to any element of A, i.e. R = φ ⊂ A × A, then the relation R in a set A is called empty relation.
Universal Relation
If each element of A is related to every element of A, i.e. R = A × A, then the relation R in set A is said to be universal relation.
Both the empty relation and the universal relation are some times 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.
Equivalence Relation- A relation R in a set A is an equivalence relation if R is reflexive, symmetric and transitive.
Functions
A function is a relationship which explains that there should be only one output for each input. It is a special kind of relation(a set of ordered pairs) which obeys a rule, i.e. every y-value should be connected to only one y-value.
Mathematically, “a relation f from a set A to a set B is said to be a function if every element of set A has one and only one image in set B”.
In other words, a function f is a relation from a set A to set B such that the domain of f is A and no two distinct ordered pairs in f have the same first element. Also, A and B are two non-empty sets.
Types of Functions
One to one Function: 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.
Onto Function: 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.
Onto function
One-one and Onto Function: A function f: X → Y is said to be one-one and onto (or bijective), if f is both one-one and onto.
One-one and Onto function
Composition of Functions
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
Invertible Functions
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.
An important note is that, 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.
Binary Operations
A binary operation ∗ on a set A is a function ∗ : A × A → A. We denote ∗ (a, b) by a ∗ b.
Example Problems
Example 1: Show that subtraction and division are not binary operations on R.
Solution: N × N → N, given by (a, b) → a – b, is not binary operation, as the image of (2, 5) under ‘–’ is 2 – 5 = – 3 ∉ N.
Similarly, ÷: N × N → N, given by (a, b) → a ÷ b is not a binary operation, as the image of (2, 5) under ÷ is 2 ÷ 5 = 2/5 ∉ N.
Example 2: 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: From the given, we have:
gof(2) = g (f(2)) = g (3) = 7
gof (3) = g (f(3)) = g (4) = 7
gof(4) = g (f(4)) = g (5) = 11
gof(5) = g (5) = 11
Example 3: Show that the relation R in the set Z of integers given by R = {(a, b) : 2 divides a – b} is an equivalence relation.
Solution: R is reflexive, as 2 divides (a – a) for all a ∈ Z.
Further, if (a, b) ∈ R, then 2 divides a – b.
Therefore, 2 divides b – a.
Hence, (b, a) ∈ R, which shows that R is symmetric.
Similarly, if (a, b) ∈ R and (b, c) ∈ R, then (a – b) and (b – c) are divisible by 2.
Now, a – c = (a – b) + (b – c) is even. (from the above statements)
From this,
(a – c) is divisible by 2.
This shows that R is transitive.
Thus, R is an equivalence relation in Z.
Answer:
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Relation: A relation R from set X to a set Y is defined as a subset of the cartesian product X × Y. We can also write it as R ⊆ {(x, y) ∈ X × Y : xRy}.
Note: If n(A) = p and n(B) = q from set A to set B, then n(A × B) = pq and number of relations = 2pq.
Types of Relation
Empty Relation: A relation R in a set X, is called an empty relation, if no element of X is related to any element of X,
i.e. R = Φ ⊂ X × X
Universal Relation: A relation R in a set X, is called universal relation, if each element of X is related to every element of X,
i.e. R = X × X
Reflexive Relation: A relation R defined on a set A is said to be reflexive, if
(x, x) ∈ R, ∀ x ∈ A or
xRx, ∀ x ∈ R
Symmetric Relation: A relation R defined on a set A is said to be symmetric, if
(x, y) ∈ R ⇒ (y, x) ∈ R, ∀ x, y ∈ A or
xRy ⇒ yRx, ∀ x, y ∈ R.
Transitive Relation: A relation R defined on a set A is said to be transitive, if
(x, y) ∈ R and (y, z) ∈ R ⇒ (x, z) ∈ R, ∀ x, y, z ∈ A
or xRy, yRz ⇒ xRz, ∀ x, y,z ∈ R.
Equivalence Relation: A relation R defined on a set A is said to be an equivalence relation if R is reflexive, symmetric and transitive.
Equivalence Classes: Given an arbitrary equivalence relation R in an arbitrary set X, R divides X into mutually disjoint subsets A, called partitions or sub-divisions 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
Ai ∪ Aj = X and Ai ∩ Aj = 0, i ≠ j. The subsets Ai and Aj are called equivalence classes.
Function: Let X and Y be two non-empty sets. A function or mapping f from X into Y written as f : X → Y is a rule by which each element x ∈ X is associated to a unique element y ∈ Y. Then, f is said to be a function from X to Y.
The elements of X are called the domain of f and the elements of Y are called the codomain of f. The image of the element of X is called the range of X which is a subset of Y.
Note: Every function is a relation but every relation is not a function.
Types of Functions
One-one Function or Injective Function: A function f : X → Y is said to be a one-one function, if the images of distinct elements of x under f are distinct, i.e. f(x1) = f(x2 ) ⇔ x1 = x2, ∀ x1, x2 ∈ X
A function which is not one-one, is known as many-one function.
Onto Function or Surjective Function: A function f : X → Y is said to be onto function or a surjective function, if every element of Y is image of some element of set X under f, i.e. for every y ∈ y, there exists an element X in x such that f(x) = y.
In other words, a function is called an onto function, if its range is equal to the codomain.
Bijective or One-one and Onto Function: A function f : X → Y is said to be a bijective function if it is both one-one and onto.
Composition of Functions: Let f : X → Y and g : Y → Z be two functions. Then, composition of functions f and g is a function from X to Z and is denoted by fog and given by (fog) (x) = f[g(x)], ∀ x ∈ X.
Note
(i) In general, fog(x) ≠ gof(x).
(ii) In general, gof is one-one implies that f is one-one and gof is onto implies that g is onto.
(iii) If f : X → Y, g : Y → Z and h : Z → S are functions, then ho(gof) = (hog)of.
Invertible Function: A function f : X → Y is said to be invertible, if there exists a function g : Y → X such that gof = Ix and fog = Iy. The function g is called inverse of function f and is denoted by f-1.
Note
(i) To prove a function invertible, one should prove that, it is both one-one or onto, i.e. bijective.
(ii) If f : X → V and g : Y → Z are two invertible functions, then gof is also invertible with (gof)-1 = f-1og-1