Relations and Functions

Relations and Functions

It is a special type of relationship (a set of pairs ordered) that complies with the law, i.e. the value of each y should be connected to the value of only one y. According to statistics, “the relationship f from set A to set B is said to be functional if every part of set A has only one image in set B”.

Lecture Notes: Relations and Functions

1. Introduction to Relations:

   • A relation is a set of ordered pairs (x, y) where x belongs to a set A and y belongs to a set B.
   • Types of relations:
     • Empty relation: A relation with no elements.
     • Universal relation: A relation where every element of A is related to every element of B.
     • Reflexive relation: A relation where every element of A is related to itself.
     • Symmetric relation: A relation where if (x, y) belongs to the relation, then (y, x) also belongs to the relation.
     • Antisymmetric relation: A relation where if (x, y) belongs to the relation and (y, x) belongs to the relation, then x = y.
     • Transitive relation: A relation where if (x, y) belongs to the relation and (y, z) belongs to the relation, then (x, z) belongs to the relation.
   • Representation of relations:
     • Arrow diagram: Representing relations using arrows between elements of sets.
     • Roster form: Listing elements of a relation inside curly braces.
     • Set-builder form: Defining a relation using set-builder notation.
     • Matrix representation: Representing a relation using a matrix.

2. Types of Relations:
   a. Reflexive Relations:
   • Reflexive relations are relations in which every element is related to itself.
   • Reflexive closure: Adding necessary ordered pairs to make a relation reflexive.
   • Reflexive relations on a set A: Relations where every element of A is related to itself.

   b. Symmetric Relations:

   • Symmetric relations are relations in which if (x, y) belongs to the relation, then (y, x) also belongs to the relation.
   • Symmetric closure: Adding necessary ordered pairs to make a relation symmetric.
   • Symmetric relations on a set A: Relations where if (x, y) belongs to the relation, then (y, x) also belongs to the relation.

   c. Transitive Relations:

   • Transitive relations are relations in which if (x, y) belongs to the relation and (y, z) belongs to the relation, then (x, z) belongs to the relation.
   • Transitive closure: Adding necessary ordered pairs to make a relation transitive.
   • Transitive relations on a set A: Relations where if (x, y) belongs to the relation and (y, z) belongs to the relation, then (x, z) belongs to the relation.

   d. Equivalence Relations:

   • Equivalence relations are reflexive, symmetric, and transitive relations.
   • Equivalence classes and partition of a set: Dividing a set into subsets called equivalence classes, where elements within the same class are related to each other.

   e. Partial Order Relations:

   • Partial order relations are reflexive, antisymmetric, and transitive relations.
   • Partially ordered sets (posets): Sets with a partial order relation.

3. Introduction to Functions:

   • A function is a relation in which each element of the domain is associated with exactly one element of the range.
   • Domain and range of a function: The set of all possible input values and output values, respectively.
   • One-to-one function: A function where each element of the domain is associated with a unique element of the range.
   • Onto function: A function where every element of the range has at least one

1. Formulas:

   a. Domain and Range of a Function:

   • Domain: Set of all possible input values of a function
   • Range: Set of all possible output values of a function

   b. Identity Function:

   • Definition: f(x) = x
   • Domain: (-∞, +∞)
   • Range: (-∞, +∞)

   c. Constant Function:

   • Definition: f(x) = k (where k is a constant)
   • Domain: (-∞, +∞)
   • Range: {k}

   d. Modulus Function:

   • Definition: f(x) = |x|
   • Domain: (-∞, +∞)
   • Range: [0, +∞)

   e. Composition of Functions:

   • Definition: (f ∘ g)(x) = f(g(x))
   • Domain: Set of all x for which g(x) is in the domain of f
   • Range: Set of all possible output values of the composite function

   f. Inverse Function:

   • Definition: f^(-1)(x)
   • Domain: Set of all y for which y is in the range of f
   • Range

• WORK SHEET 1
• WORK SHEET 2
• WORK SHEET 3
• MCQ WORK SHEET 1
• MCQ WORK SHEET 2
• MCQ WORK SHEET 3
• CHAPTER TEST 1
• CHAPTER TEST 2
• CHAPTER TEST 3