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