Relation and function

• Relations - A relation R from a non-empty 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.
• Functions - 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.
• Please note that all functions are relations but all relations are not functions.
• Relations and functions can be represented in different forms such as arrow representation, algebraic form, set-builder form, graphical form, roster form, and tabular form.
• Set-builder form - {(x, y): y = x2, x ∈ A, y ∈ B}
• Roster form - {(1, 1), (2, 4), (3, 9)}
• Arrow representation

  
• Table Representation

  
• Relation - a single input may have multiple outputs.
• Function - input has a single output.
• A relation in math is a set of ordered pairs defining the relation between two sets.
• A function is a relation in math such that each element of the domain is related to a single element in the codomain.
• A relation may or may not be a function.
• All functions are relations.
• Empty Relation - if it has no elements, that is, no element of set A is mapped or linked to any element of A. It is denoted by R = ∅.
• Universal Relation - A relation R in a set A is a universal relation if each element of A is related to every element of A, i.e., R = A × A. It is called the full relation.
• Identity Relation - A relation R on A is said to be an identity relation if each element of A is related to itself, that is, R = {(a, a) : for all a ∈ A}
• Inverse Relation - Define R to be a relation from set P to set Q i.e., R ∈ P × Q. The relation R-1 is said to be an Inverse relation if R-1 from set Q to P is denoted by R-1 = {(q, p): (p, q) ∈ R}.
• Reflexive Relation - A binary relation R defined on a set A is said to be reflexive if, for every element a ∈ A, we have aRa, that is, (a, a) ∈ R.
• Symmetric Relation - A binary relation R defined on a set A is said to be symmetric if and only if, for elements a, b ∈ A, we have aRb, that is, (a, b) ∈ R, then we must have bRa, that is, (b, a) ∈ R.
• Transitive Relation - A relation R is transitive if and only if (a, b) ∈ R and (b, c) ∈ R ⇒ (a, c) ∈ R for a, b, c ∈ A
• Equivalence Relation - A relation R defined on a set A is said to be an equivalence relation if and only if it is reflexive, symmetric and transitive
• Antisymmetric Relation - A relation R on a set A is said to be antisymmetric if (a, b) ∈ R and (b , a) ∈ R ⇒ a = b.
• One-to-One Function - A function f: A → B is said to be one-to-one if each element of A is mapped to a distinct element of B. It is also known as Injective Function.
• Onto Function - A function f: A → B is said to be onto, if every element of B is the image of some element of A under f, i.e, for every b ∈ B, there exists an element a in A such that f(a) = b. A function is onto if and only if the range of the function = B.
• Many to One Function - A many to one function is defined by the function f: A → B, such that more than one element of the set A are connected to the same element in the set B.
• Bijective Function - A function that is both one-to-one and onto function is called a bijective function.
• Constant Function - The constant function is of the form f(x) = K, where K is a real number. For the different values of the domain(x value), the same range value of K is obtained for a constant function.
• Identity Function - An identity function is a function where each element in a set B gives the image of itself as the same element i.e., g (b) = b ∀ b ∈ B. Thus, it is of the form g(x) = x.
• Algebraic functions are based on the degree of the algebraic expression. The important algebraic functions are:

  • linear function
  • quadratic function
  • cubic function
  • polynomial function
  • objective functions