**Relation:** A relation R from set X to a set Y is defined as a subset of the cartesian product X × Y. We can also write it as R ⊆ {(x, y) ∈ X × Y : xRy}.

Note: If n(A) = p and n(B) = q from set A to set B, then n(A × B) = pq and number of relations = 2^{pq}.

**Types of Relation**

**Empty Relation:** A relation R in a set X, is called an empty relation, if no element of X is related to any element of X,

i.e. R = Φ ⊂ X × X

**Universal Relation:** A relation R in a set X, is called universal relation, if each element of X is related to every element of X,

i.e. R = X × X

**Reflexive Relation:** A relation R defined on a set A is said to be reflexive, if

(x, x) ∈ R, ∀ x ∈ A or

xRx, ∀ x ∈ R

**Symmetric Relation:** A relation R defined on a set A is said to be symmetric, if

(x, y) ∈ R ⇒ (y, x) ∈ R, ∀ x, y ∈ A or

xRy ⇒ yRx, ∀ x, y ∈ R.

**Transitive Relation:** A relation R defined on a set A is said to be transitive, if

(x, y) ∈ R and (y, z) ∈ R ⇒ (x, z) ∈ R, ∀ x, y, z ∈ A

or xRy, yRz ⇒ xRz, ∀ x, y,z ∈ R.

**Equivalence Relation:** A relation R defined on a set A is said to be an equivalence relation if R is reflexive, symmetric and transitive.

**Equivalence Classes:** Given an arbitrary equivalence relation R in an arbitrary set X, R divides X into mutually disjoint subsets A, called partitions or sub-divisions of X satisfying

- all elements of A
_{i}are related to each other, for all i. - no element of A
_{i}is related to any element of A_{j}, i ≠ j - A
_{i }∪ A_{j}= X and A_{i}∩ A_{j}= 0, i ≠ j. The subsets A_{i}and A_{j}are called equivalence classes.

**Function:** Let X and Y be two non-empty sets. A function or mapping f from X into Y written as f : X → Y is a rule by which each element x ∈ X is associated to a unique element y ∈ Y. Then, f is said to be a function from X to Y.

The elements of X are called the domain of f and the elements of Y are called the codomain of f. The image of the element of X is called the range of X which is a subset of Y.

Note: Every function is a relation but every relation is not a function.

**Types of Functions**

**One-one Function or Injective Function:** A function f : X → Y is said to be a one-one function, if the images of distinct elements of x under f are distinct, i.e. f(x_{1}) = f(x_{2} ) ⇔ x_{1} = x_{2}, ∀ x_{1}, x_{2} ∈ X

A function which is not one-one, is known as many-one function.

**Onto Function or Surjective Function:** A function f : X → Y is said to be onto function or a surjective function, if every element of Y is image of some element of set X under f, i.e. for every y ∈ y, there exists an element X in x such that f(x) = y.

In other words, a function is called an onto function, if its range is equal to the codomain.

**Bijective or One-one and Onto Function:** A function f : X → Y is said to be a bijective function if it is both one-one and onto.

**Composition of Functions:** Let f : X → Y and g : Y → Z be two functions. Then, composition of functions f and g is a function from X to Z and is denoted by fog and given by (fog) (x) = f[g(x)], ∀ x ∈ X.

Note

(i) In general, fog(x) ≠ gof(x).

(ii) In general, gof is one-one implies that f is one-one and gof is onto implies that g is onto.

(iii) If f : X → Y, g : Y → Z and h : Z → S are functions, then ho(gof) = (hog)of.

**Invertible Function:** A function f : X → Y is said to be invertible, if there exists a function g : Y → X such that gof = I_{x} and fog = I_{y}. The function g is called inverse of function f and is denoted by f^{-1}.

Note

(i) To prove a function invertible, one should prove that, it is both one-one or onto, i.e. bijective.

(ii) If f : X → V and g : Y → Z are two invertible functions, then gof is also invertible with (gof)^{-1} = f^{-1}og^{-1}

**Domain and Range of Some Useful Functions**

**Binary Operation:** A binary operation * on set X is a function * : X × X → X. It is denoted by a * b.

**Commutative Binary Operation:** A binary operation * on set X is said to be commutative, if a * b = b * a, ∀ a, b ∈ X.

**Associative Binary Operation:** A binary operation * on set X is said to be associative, if a * (b * c) = (a * b) * c, ∀ a, b, c ∈ X.

Note: For a binary operation, we can neglect the bracket in an associative property. But in the absence of associative property, we cannot neglect the bracket.

**Identity Element:** An element e ∈ X is said to be the identity element of a binary operation * on set X, if a * e = e * a = a, ∀ a ∈ X. Identity element is unique.

Note: Zero is an identity for the addition operation on R and one is an identity for the multiplication operation on R.

**Invertible Element or Inverse:** Let * : X × X → X be a binary operation and let e ∈ X be its identity element. An element a ∈ X is said to be invertible with respect to the operation *, if there exists an element b ∈ X such that a * b = b * a = e, ∀ b ∈ X. Element b is called inverse of element a and is denoted by a^{-1}.

Note: Inverse of an element, if it exists, is unique.

**Operation Table:** When the number of elements in a set is small, then we can express a binary operation on the set through a table, called the operation table.