Relation:

Relation and function have a wide importance in mathematics.

A relation R from a non-empty set A to a non-empty set B is a subset of A × B.
The subset is simply the converge attachment of first element and the second element of the ordered pairs in A × B.

Terminology related to Relations:

1. Ordered pair: A pair of elements when grouped together in a particular order is said to be ordered pair.
2. Cartesian Products of Sets: The set of all the orderd pairs of elements from one set to other is said to be Cartesian product of sets.
   Again let us consider two sets A and B. The cartesian product A × B is the set of all ordered pairs of elements from A and B, i.e., A × B = { (a,b) : a ∈ A, b ∈ B }.
   If anyone of A and B is the null set, then A × B will also be empty set, i.e., A × B = φ

Rules & Formula:

1. Given, If (a, b) = (x, y), then a = x and b = y.
2. If n(A) = p and n(B) = q then n(AxB) = pq
3. If either n(A) = ∞ or n(B) = ∞ then n(AxB) is ∞.
4. A × A × A = {(a, b, c) : a, b, c ∈ A}. Here (a, b, c) is called an ordered triplet.
5. In general, A × B ≠ B × A.
6. Domain: 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.
7. Co-domain: 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 co-domain of the relation R.
   range ⊂ co-domain.

Types of Relations:

Empty relation: If there is no any element of A that is related to any element of A, i.e., R = φ ⊂ A × A. Then this relation R is called empty relation.

Universal relation: If each element of A is related to every element of A, i.e., R = A × A. Then we call the relation R in that set A as universal relation.

Both the types of relationships are sometimes trivial relations.

Reflexive relation: When the Same element is present as co-domain or simply R in X is a relation with (a, a) ∈ R ∀ a ∈ X.

All these constitute in the study of relation and function.

Base types:

Symmetric relation: A relation R in X is a relation satisfying (a, b) ∈ R implies (b, a) ∈ R.

Transitive relation: A relation R in X is a relation satisfying (a, b) ∈ R and (b, c) ∈ R implies that (a, c) ∈ R.

Equivalence relation: A relation R in X is a relation which is reflexive, symmetric and transitive.

Equivalence class:  [a] containing a ∈ X for an equivalence relation R in X is the subset of X containing all elements b related to a.

Relation and function go hand in hand.

Function:

A function f from a set A to a set B is a specific type of relation for which every element x of set A has one and only one image y in set B. i.e each element in A has a unique element in B.

We symbolize any function as f: A→B, where f(x) = y where A is the domain and B is the codomain of “f”. Range of function is the set of images of each element in domain.

Algebra of functions:

For functions f : X → R and g : X → R, we have

1. (f + g) (x) = f (x) + g(x), x ∈ X
2. (f – g) (x) = f (x) – g(x), x ∈ X
3. (f.g) (x) = f (x) .g (x), x ∈ X
4. (kf) (x) = k ( f (x) ), x ∈ X, where k is a real number.
5. \(\large{(}\frac{f}{g})\)(x) = \(\frac{f(x)}{g(x)})\) , x ∈ X, g(x) ≠ 0
   

Also learn Algebra formulas.

Properties for relation and function:

One-one Function: We can say a function f : X → Y as one-one (or injective) if f(x₁) = f(x₂) ⇒ x₁ = x₂ ∀ x₁ , x₂ ∈ X.

On-to function (surjective function): We can say a function f : X → Y as onto (or surjective) if given any y ∈ Y, ∃ x ∈ X such that f(x) = y.

One-one and onto (or bijective): We can say a function f : X → Y as one-one and onto (or bijective), if f is both one-one and onto.

Composition of functions: The composition of functions f : A → B and g : B → C is the function with symbol as gof : A → C and actually is gof(x) = g(f(x)) ∀ x ∈ A.

Invertible function: A function f : X → Y is invertible if ∃ g : Y → X such that gof = I_X and fog = I_Y. Also, A function f : X → Y is invertible if and only if f is one-one and onto.

Binary Operation on relation and function:

Its symbol is *. It is a function * from A X A to A.

Identity element: As earlier said that element e ∈ X is the identity element for binary operation ∗ : X × X → X, if a ∗ e = a = e ∗ a ∀ a ∈ X.

An element a ∈ X is invertible for binary operation ∗ : A × A → X, if there exists b ∈ A such that a ∗ b = e = b ∗ a where e is the identity for the binary operation ∗. The element b is inverse of a and is we denote it by a^–1.

Commutative Operation: If a ∗ b = b ∗ a ∀ a, b in X then operation ∗ on X is commutative.

Associative Operation: If (a ∗ b) ∗ c = a ∗ (b ∗ c) ∀ a, b, c in X than the operation ∗ on X is associative.

Relation and function Examples:

1. Let A = {1,2,3}, B = {3,4} and C = {4,5,6}. Find A × (B ∩ C).

Solution:

We know that by the definition of the intersection of two sets, (B ∩ C) = {4}.

Therefore, A × (B ∩ C) = {(1,4), (2,4), (3,4)}.

2. If P = {1, 2}, form the set P × P × P.

Solution:

We can simply write its 3 different element in a ordered triplet

P × P × P = {(1,1,1), (1,1,2), (1,2,1), (1,2,2), (2,1,1), (2,1,2), (2,2,1),(2,2,2)}.

3. If A × B ={(p, q),(p, r), (m, q), (m, r)}, find A and B.

Solution:

A = set of first elements = {p, m}

B = set of second elements = {q, r}.

4. Let f = {(1,1), (2,3), (0, –1), (–1, –3)} be a linear function from Z into Z. Find f(x).

Solution:
we know that f is a linear function, so we can write f (x) = mx + c.

Also, since (1, 1), (0, – 1) ∈ R. f (1) = m + c = 1 and f (0) = c = –1.

This gives m = 2 and f(x) = 2x – 1.

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