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 $\mathrm{A}\stackrel{f}{\to }\mathrm{B}$

Which reads f maps A to B or f is a function from A to B.

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 $R\subseteq A\times B$.

Total number of relations if n(A) = m and n(B) = n is 2^mn. 

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:

1. Universal Relations
2. Void Relations
3. Identity Relations
4. Reflexive Relations
5. Symmetric Relations
6. Transitive Relations
7. Equivalence Relations

Let us discuss all types of relations one by one.

1. 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.

2. Void Relations

A relation R on set A is called Void/ Empty Relation if no element of A is related to any element of A.

i.e., $R\subseteq A\times B.$

3. Identity Relations

A relation 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}.

4. Reflexive Relations

A relation 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.

5. Symmetric Relations

A relation 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.

6. 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.

7. 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

1. Identity Function
2. Constant Function
3. Polynomial Function
4. Modulus Function
5. Signum Function
6. Greatest Integer Function

Let us discuss these functions in detail

1. 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.

2. 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}.

3. Polynomial Function

A polynomial function f: R → R is defined such that for each x in R, y = f(x) = a₀ + a₁x + a₂x²+ …. + a_nxⁿ, where n represents a non-negative integer, while a₀, a₁, a₂ , … , a_n are real numbers.

4. 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)=$\{\begin{array}{c}x,x\ge 0\\ -x,x<0\end{array}$

5. Signum Function

The function f:  R → R is defined by

$f(x)=\{\begin{array}{l}1,\text{ if }x>0\\ 0,\text{ if }x=0\\ -1,\text{ if }x<0\end{array}$

$f(x)=\{\begin{array}{l}1,\text{ if }x>0\\ 0,\text{ if }x=0\\ -1,\text{ if }x<0\end{array}$

Is called the signum function.

6. Greatest Integer Function

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.

$[\mathrm{x}]=\{\begin{array}{ll}-1\text{ for }& -1\le \mathrm{x}<0\\ 0\text{ for }& 0\le \mathrm{x}<1\\ 1\text{ for }& 1\le \mathrm{x}<2\\ 2\text{ for }& 2\le \mathrm{x}<3\dots \dots .\end{array}$

and so on.

6.0Difference Between Relation and Function

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(x_2​) ⇒ 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 $f(x)=\frac{1}{\sqrt{|[|x|-5]|-11}}$ where [.] denotes greatest integer function.

Solution: 

[|x|-5]|>11

So, [|x|-5]>11 or [|x|-5]<-11

$\Rightarrow [|x|]>16$ [|x|]<-6

$\Rightarrow |x|\ge 17$ |x|<-6 (Not Possible)

⇒ x ≤ -17 or x ≥ 17

So, x ∈ (–∞, –17] ∪ [17, ∞)