Revision: Section A >> Relations and Functions Mathematics ISC (Commerce) Class 12 CISCE

A relation R from a non-empty set A to a non-empty set B is a subset of the cartesian product  A × B. The subset is derived by describing a relationship between the first element and the second element of the ordered pairs in A × B. The second element is called the image of  the first element.

The set of all first elements of the ordered pairs in a relation R from a set A to a set B is called the domain of the relation R.

The set of all second elements in a relation R from a set A to a set B is called the range of the relation R. The whole set B is called the codomain of the relation R. Note that range ⊂ codomain.

Given two non-empty sets P and Q. The cartesian product P × Q is the set of all ordered pairs of elements from P and Q, i.e., P × Q = { (p,q) : p  ∈ P, q  ∈ Q } If either P or Q is the null set, then P × Q will also be empty set, i.e., P × Q = Ø

A relation R on a set A is said to be symmetric if

(a, b) ∈ R

⇒  (b, a)  ∈ R for all a b ∈ A

i.e.  aRb ⇒ bRa for all a , b ∈ A

A relation R on a set A is said to be transitive if

(a, b) ∈ R and (b, c) ∈ R

⇒ (a, c) ∈ R for all a, b , c ∈ R

i.e. aRb and bRc

⇒ aRc for all a, b, c ∈ R

A relation R on set A is said to be an equivalence relation if
(i) it is reflexive,
(ii) it is symmetric and
(iii) it is transitive.

Relation R on set A satisfying all the above three properties is an equivalence relation.

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.

A function which has either R or one of its subsets as its range is called a real valued function. Further, if its domain is also either R or a subset of R, it is called a real function.

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

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.

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.

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.

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.

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.

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.

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

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.

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 \[2^{n^2-n}\]​

Types of Function: 

Special Types of Functions:

(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 ∘$IA=f$

• 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 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 = \[n^{n^2}\]