In the context of Set Theory, which term describes a set that contains all possible subsets of a given original set, including the empty set?
1.The Subset
2. The Complement of the Set
3.The Power Set
4.The Universal Set
Frequently Asked Questions
Relation is a set of ordered pairs where the first element (input) is related to the second element (output) based on a rule or condition. A function is a special type of relation where each input value (domain element) is associated with exactly one output value (range element).
The application of the vertical line test allows for the determination of whether a relation qualifies as a function.
In a function f: A → B, the domain is the set of all possible input values (elements of set A) for which the function is defined. It represents the values that can be plugged into the function to obtain valid outputs. The range of the function is the set of all possible output values (elements of set B) that the function can produce based on its domain. It represents the values that the function can yield as results.
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Relations and Functions
Relation:
The relation is a linear operation that establishes a relationship between the elements of two sets according to some definite rule of relationship.
R: {(a, b)| (a, b) ∈ A × B and a R b}
Function:
Let A and B be two sets, and let a rule or manner or correspondence ‘f’ exist that associates each element of A with a unique element in B. Then, f is called a mapping or function from A to B.
f: A → B or AfB
Which reads f maps A to B or f is a function from A to B.
Related Video:
1.0Definition of Relation
A Relation R from set A to set B is established as a subset of the Cartesian product A × B. In other words, relation R consists of ordered pairs where the first element belongs to set A and the second element belongs to set B. Therefore, we can say that R is a relation from A to B, represented as R⊆A×B.
Total number of relations if n(A) = m and n(B) = n is 2mn.
2.0Domain and Range of a Relation
Let R be a relation from a set A to set B. The domain of R comprises all the first elements of the ordered pairs in R.
The Range of R is the set of all second elements of the ordered pairs.
Thus, Domain = {a: (a, b) ∈ R}
and Range= {b: (a, b) ∈ R}
It is clear from the above definition that the domain of a relation from set A to B is a subset of A, and its Range is a subset of B.
3.0Types of Relations
Different types of relations are:
Universal Relations
Void Relations
Identity Relations
Reflexive Relations
Symmetric Relations
Transitive Relations
Equivalence Relations
Let us discuss all types of relations one by one.
Universal Relations
A relation R on set A is termed a Universal Relation if every element of A is related to every other element within set A.
i.e., R = A × A.
Void Relations
Arelation R on set A is called Void/ Empty Relation if no element of A is related to any element of A.
i.e., R⊆A×B.
Identity Relations
Arelation defined on a set A is called an Identity Relation if each element of A is related to only and only itself.
i.e., I = {(a, a): ∀ a ∈ A}.
Reflexive Relations
Arelation defined on a set A is called Reflexive Relation if each element of A is related to itself.
I.e., (a, a) ∈ R ∀ a ∈ A.
Symmetric Relations
Arelation defined on a set A is said to be Symmetric Relation if
(a, b) ∈ R ⇒ (b, a) ∈ R for all a, b ∈ A.
i.e., a R b ⇒ b R a ∀ a, b, ∈ A.
Transitive Relations
Let A be any set. A relation R on set A is said to be a Transitive Relation if
(a, b) ∈ R and (b, c) ∈ R ⇒ (a, c) ∈ R for all a, b, c ∈ A.
i.e., a R b, b R c ⇒ a R c ∀ a, b, c ∈ A.
Equivalence Relations
A relation R on a set A is said to be an Equivalence Relation if:
It is Reflexive Relation
It is Symmetric Relation
It is Transitive Relation
4.0Function
A function f from set A to set B is defined as a relation where each element in A maps to exactly one element in B. This means that for every input from A, B has a unique output. We can represent this function as f: A → B. If (a, b) ∈ f, then f(a) = b, where b represents the image of a under f, while a is referred to as the preimage of b under f.
5.0Types of Function
Identity Function
Constant Function
Polynomial Function
Modulus Function
Signum Function
Greatest Integer Function
Let us discuss these functions in detail
Identity Function
Consider the set R of real numbers. Define the real-valued function f: R → R as y = f(x) = x for all x ∈ R. This type of function is known as the identity function, denoted by f(x) = x. Both domain and range of f are R. Its graph is a straight line that passes through the origin.
Constant Function
The function f: R → R can be expressed as y = f(x) = c for all x ∈ R, where c is a constant. In this function, the domain of f comprises all real numbers, denoted as R, and its range is the singleton set {c}.
Polynomial Function
A polynomial function f: R → R is defined such that for each x in R, y = f(x) = a0 + a1x + a2x2+ …. + anxn, where n represents a non-negative integer, while a0, a1, a2 , … , an are real numbers.
Modulus Function
The modulus function f: R → R, defined as f(x) = |x| ∀ x ∈ R, is called the absolute value function. When x is non-negative, f(x) equals x, but when x is negative, f(x) becomes the negative of x, i.e., f(x)={x,x≥0−x,x<0
The function f: R → R defined as f(x) = [x], where x ∈ R, represents the greatest integer function. Here, the notation [x] denotes the greatest integer less than or equal to x.
[x]=⎩⎨⎧−1 for 0 for 1 for 2 for −1≤x<00≤x<11≤x<22≤x<3…….
and so on.
6.0Difference Between Relation and Function
Differentiating /Parameter
Relation
Function
Definition
A relation is a connection between sets of values, often represented as a subset of the Cartesian product.
A function is a relation in which each input is associated with exactly one output.
Denotation
The symbol "R" is used to denote a relation.
A function is represented by the symbols "F" or "f".
Example
R = {(p, 1), (p, 2), (q, 1)}
F = {(p, 1), (q, 2)}
Note
Not every relation is a function.
Every function is a type of relation.
7.0Relation and Function Formulas (R: A → B)
Relation (R): R ⊆ A × B
Domain (D): D = { x ∣ (x, y) ∈ R}
Codomain (C): C = B
Range (R): R = { y ∣ ∃ x ∈ D, (x, y) ∈ R}
Cartesian Product (A × B): A × B = {(a, b) ∣ a ∈ A and b ∈ B}
Vertical Line Test: The vertical line test confirms that a relation is a function if each vertical line crosses the graph at most once.
One-to-One Function: f(x1) = f(x2) ⇒ x1 = x2 only
Onto Function (Surjective): Every element in the codomain has at least one preimage.
Bijective: Both one-to-one and onto.
Composition of Functions (f o g): (fog)(x) = f( g(x))
8.0Relations and Functions Examples
Question 1. In the set N × N consider the relation R defined as (a, b) R (c, d) ⇔ad = bc; a, b, c, d ∈ N
Show that this is an equivalence relation.
Solution:
For proving the relation R on N × N to be an equivalence relation, we will prove that the relation R is reflexive, symmetric and transitive.
Reflexive:
Let x, y ∈ N, then
⇒ (x, y) ∈ N × N ⇒ xy = yx, by commutativity in N
⇒ (x, y) R (x, y) ∀ (x, y) ∈ N × N
∴ R is reflexive.
Symmetric:
Let (x, y) R (p, q) where x, y, p, q ∈ N, then
(x, y) R (p, q) ⇒ xq = yp ⇒ yp = xq
⇒ py = qx[by commutativity in N]
⇒ (p, q) R (x, y)
∴R is symmetric.
Transitive:
Let (x1, y1) R (x2, y2) and (x2, y2) R (x3, y3) where
⇒ (x1, y1) (x2, y2), (x3, y3) ∈ N × N, then
⇒ (x1, y1) R (x2, y2) ⇒ x1 y2 = y1 x2...(1)
∴ (x2, y2) R (x3, y3) ⇒ x2 y3 = y2 x3...(2)
putting the value of x2 from (2) into (1)
⇒ x1 y3 = y1 x3...(3)
⇒ (x1, y1) R (x3, y3)
Therefore, R is transitive.
Hence the relation defined on N × N is an equivalence relation.
Question 2. Find the domain of function y=\sqrt{5-2 x} .
Solution:
5 – 2x > 0 ⇒ x \leq \frac{5}{2}
∴ Domain is (–∞, 5/2]
Question 3. Find the range of function f(x) = 3 – cos x.
Solution:
f(x) = 3 – cos x
⇒ –1 ≤ cos x ≤ 1
⇒2 ≤ 3 – cos x ≤ 4
∴ Range of f(x) = [2, 4]
Question 4. Find the domain f(x)=∣[∣x∣−5]∣−111 where [.] denotes greatest integer function.
