Relations and Functions Quiz
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Questions and Answers

What is a relation from a set A to a set B?

  • A subset of B
  • A subset of A
  • A subset of B × A
  • A subset of A × B (correct)

    • What is the domain of a relation?

  • The set of all elements in B
  • The set of all second elements of the ordered pairs
  • The set of all elements in A
  • The set of all first elements of the ordered pairs (correct)

    • What is the range of a relation?

  • The set of all elements in A
  • The set of all second elements of the ordered pairs (correct)
  • The set of all first elements of the ordered pairs
  • The set of all elements in B

    • What is a function from a set A to a set B?

    <p>A relation from A to B with every element of A related to exactly one element of B</p> Signup and view all the answers

    What is the domain of a function?

    <p>The set of all elements in A</p> Signup and view all the answers

    What is the codomain of a function?

    <p>The set of all elements in B</p> Signup and view all the answers

    What is an one-one function?

    <p>A function where different elements of the domain have different images in the codomain</p> Signup and view all the answers

    What is an onto function?

    <p>A function where every element of the codomain is the image of some element in the domain</p> Signup and view all the answers

    What is a bijective function?

    <p>A function that is both one-one and onto</p> Signup and view all the answers

    What is the range of a function?

    <p>The set of all elements in B that are related to some element in A</p> Signup and view all the answers

    Study Notes

    Relations and Functions

    Relations

    • A relation from a set A to a set B is a subset of A × B.
    • It can be represented as:
      • R = {(a, b) | a ∈ A, b ∈ B, and a is related to b}
      • Example: Let A = {1, 2, 3} and B = {4, 5}. Then, {(1, 4), (2, 5), (3, 4)} is a relation from A to B.
    • Domain of a relation: The set of all first elements of the ordered pairs in the relation.
    • Range of a relation: The set of all second elements of the ordered pairs in the relation.

    Functions

    • A function is a relation from a set A to a set B such that:
      • Every element of A is related to exactly one element of B.
      • A function is denoted as f: A → B, where f is the function and A and B are the domain and codomain respectively.
    • Domain of a function: The set of all elements in A.
    • Range of a function: The set of all elements in B that are related to some element in A.
    • Codomain of a function: The set of all elements in B.

    Types of Functions

    • One-One Function (Injective): A function is one-one if different elements of the domain have different images in the codomain.
    • Onto Function (Surjective): A function is onto if every element of the codomain is the image of some element in the domain.
    • One-One Correspondence (Bijective): A function is bijective if it is both one-one and onto.

    Inverse of a Function

    • If f: A → B is a bijective function, then there exists a function g: B → A such that:
      • f(g(b)) = b for all b ∈ B
      • g(f(a)) = a for all a ∈ A
    • The function g is called the inverse of f, denoted as f^(-1).

    Composite Functions

    • If f: A → B and g: B → C are two functions, then the composite function gof: A → C is defined as:
      • (gof)(a) = g(f(a)) for all a ∈ A

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    Description

    Test your knowledge of relations and functions, including domain, range, and types of functions such as one-one, onto, and bijective functions. Learn about inverse and composite functions too!

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