Types of Relations
* A relation R from a non-empty set A to another non-empty set B is the subset of A× B.
* The subset, R, is derived by describing a relationship between the first element and the second element of the ordered pairs in A × B.
* If n(A) = p, n(B) = q; then n(A × B) = pq and the total number of possible relations from set A to set B is 2^{pq}.
* A relation R in a set A is called an empty relation, if no element of A is related to any element of A, i.e.
R = ∅ ⊂ A × A.
* The relation R in n set A is called a universal relation or a total relation, if every element of A is related to every element of A, i.e.
R = A × A.
Let R be a relation in T. If (x, x) ∈ R for every x ∈ T, then R is a reflexive relation.
Let R be a relation in a set T. If (x, x) ∈ R implies that (y, x) ∈ R ∀ x, y ∈ T, then R is a symmetric relation.
* A relation R, in T is said to be transitive, if (x, y) ∈ R and (y, z) ∈ R implies that
(x, y) ∈ R ∀ x, y, z ∈ T.
* A relation is said to be equivalence relation if it is reflexive, symmetric and transitive.
Types of Functions
* A function, ƒ: A → B, is defined to be one-one, if the images of the distinct elements of A under ƒ are distinct, i.e. for every x_{1} , x_{2}
∈ A, ƒ (x_{1} ) = ƒ ( x_{2} )
implies x_{1} = x_{2}.
* A one-one function is also known as an injective function or simply an injection.
* If a function is not one-one, then it is known as many-one.
* The horizontal line test helps us to check the injectivity of a function.
* A function, ƒ from a set A to a set B, is said to be onto, if and only if for every element y or B, there is an element x in A such that ƒ(x) = y. Or ƒ is onto if and
only if ƒ(A) = B.
* Onto functions are also known as surjective functions or simply surjections.
Bijective Functions
* A function, ƒ: A →B, is a bijection if it is both one-one and onto.
* A one-one function, ƒ: X → X, is necessarily onto, for every finite set X.
* An onto function, ƒ: X → X, is necessarily one-one, for every finite set X.
Composition of Functions
* Let ƒ: A → B and ɡ: B → C be two functions. Then the composition of ƒ and ɡ, denoted by ɡo ƒ is defined as the function ɡo ƒ: A → C given by ɡo ƒ (x) = ɡ (ƒ(x)),
∀ x ∈ A.
* The function ɡo ƒ is possible only if the range of the function ƒ is the domain of the function ɡ.
* When ɡo ƒ is possible, ƒ oɡ may or may not be possible.
* If both ɡo ƒ and ƒ oɡ are possible, then they may or may not be equal.
Theorem on Composition of Functions
* If ƒ: A → B and ɡ: B → C are one-one, then ɡo ƒ: A →C is also one-one.
* If ƒ: A → B and ɡ: B → C are onto, then ɡo ƒ: A →C is also onto.
Composition of functions satisfies the associative property.
Inverse of a Bijective Functions
* Let ƒ: A → B be a function. If for an arbitrary x ∈ A we have ƒ(x) = y in B, then function, ɡ: B→ A given by ɡ(y) = x, where y ∈ B and x ∈ A, is called the inverse
function of ƒ.
* A function ƒ: A → B, is said to be invertible if there exists a function, ɡ: B→ A, such that ɡo ƒ = I_{A} and ƒ oɡ = I_{B}.
The function, ɡ, is called the inverse of ƒ and is denoted by ƒ^{-1}.
* The composition of two invertible functions is also invertible.
* If ƒ and ɡ are two invertible functions, then (goƒ)^{-1} = ƒ^{-1} og^{-1}.
Binary Operations
* A binary operation * on a set, A, is a function *: A × A →A. * (a, b) is denoted by a * b.
* In a binary operation, a pair of elements from A associates and results into a single elements of A.
* Operation table: when the number of elements in a set, A, is small, we can express a binary operation * on set a through a table called the operation table for the
operation,’* ’.
* A binary operation ∗ on a set, A, is called commutative, if a ∗ b = b ∗ a, for every a, b ∈ A.
* A binary operation ∗: A × A → A is said to be associative if (a ∗ b) ∗ c = a ∗ (b ∗ c), ∀ a, b, c, ∈ A.
* Identity: Given a binary operation ∗: A × A → A, an element e ∈ A, if it exists, is called the identity for the operation ∗, a ∗ e = a = e ∗ a, ∀ a ∈ A.
* Inverse: : Given a binary operation ∗ : A × A → A with the identity element, e, in A, an element a ∈ A is said to be invertible with respect to the operation ∗, if there
exits an element, b, in a such that
a ∗ b = e = b ∗ a. where is called the inverse of a and is denoted by a^{-1}.