Overview of Relations and Functions

An ordered pair is a pair of objects in which the order of the objects is important.

It is written as (a, b), where a is called the first component and b is called the second component.

Note:
In general,

(a,b) ≠ (b,a)

i.e., the order of elements matters.

If A and B are two sets, then the Cartesian product of A and B, denoted by A × B, is the set of all ordered pairs (a, b) such that a ∈ A and b ∈ B.

A × B ={(a,b):a ∈ A, b ∈ B}

Result:
If n(A) = m and n(B) = n, then

n(A × B) = m × n

A relation is a set of ordered pairs.

The set of all first elements of the ordered pairs is called the domain, and the set of all second elements that appear is called the range.

Types of Relations:

Special Types of Relations:

Important Result

If a set A contains n elements, then the number of reflexive relations on A is \[2^{n^2-n}\]​

Equivalence Relation:
A relation R on a set A is an equivalence relation if it is
Reflexive, Symmetric and Transitive.

Important Result:

• Equality → equivalence relation

• “is similar to” (triangles) → equivalence relation

• “is perpendicular to” → not an equivalence relation

Equivalence Class:
If R is an equivalence relation on A and a ∈ A, then

[a] ={x ∈ A : (x, a) ∈ R}.

Properties:

1. Every element belongs to exactly one equivalence class.

2. Distinct equivalence classes are disjoint.

3. The union of all equivalence classes is A.

Let m be a positive integer and x, y ∈ Z

Then x is said to be congruent to y modulo m, written as

x ≡ y (mod m)

iff x − y is divisible by m.

A function f: X→Y is a relation such that:

1. Every element of X has an image in Y

2. Each element of X has exactly one image in Y

Domain:
The domain of a function f is the set of all elements of X for which the function f is defined.

Co-domain:

The codomain of a function f is the set Y into which the function maps elements of the domain.

Range:

The range of a function f is the set of all images of elements of the domain under the function f.

Types of Function: 

Special Types of Functions:

Let A, B, and C be three non-empty sets.

If f: A→B and g: B→C are two functions, then the composition of f and g, denoted by

(g∘f)(x) = g(f(x)),for all x ∈ A.

The composite function is also called the resultant function or function of a function.

(g∘f)(x) = g(f(x))

• Exists only if Range of f ⊆ Domain of g

• Order matters: g∘f ≠ f∘g

• Associative: h∘(g∘f) = (h∘g)∘f

• If f and g are one-one, then g∘f is one-one

• If f and g are onto, then g∘f is onto

• Identity property:
  I_B∘f = f . f ∘$IA=f$

A function f: X→Y is called invertible if it is one-one and onto.
In this case, there exists a function f^−1:Y→X such that

f⁻¹(y) = x  ⟺  f(x) = y

The function f⁻¹ is called the inverse of f.

• Only bijective (one-one/onto) functions are invertible.

• Domain of f⁻¹ = Range of f and Range of f⁻¹ Domain of f.

• f⁻¹(y) = x if and only if f(x) = y.

• f⁻¹∘f = I_X ​ and f∘f⁻¹ = I_Y​.

• The inverse of a bijective function is unique and (f⁻¹)⁻¹ = f

A binary operation (or composition) on a non-empty set A is a function

∗ : A × A → A

Which associates each ordered pair (a,b) in A×A with a unique element a ∗ b in A.

• A binary operation must satisfy closure, i.e.
  a, b ∈ A ⇒ a ∗ b ∈ A
• Order matters in a binary operation; in general,
  a ∗ b ≠ b ∗ a

• Addition and multiplication are binary operations on N, Z, Q, R.

• Subtraction and division are not binary operations on N.

• Division is a binary operation on R−{0}.

• Union and intersection are binary operations on the power set P(A).

• If a finite set A contains n elements, then
  Number of binary operations on A = \[n^{n^2}\]