Relations and Functions in Mathematics
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Questions and Answers

What is the Cartesian product of two non-empty sets A and B?

  • The set of all possible subsets of A and B
  • The union of sets A and B
  • The set of all elements common to A and B
  • The set of all possible ordered pairs of elements from A and B (correct)

How many relations can exist between set A with m elements and set B with n elements?

  • n^m
  • m^n
  • 2^(n^m)
  • 2^(m*n) (correct)

Which statement is true for a relation to be classified as a function?

  • No element in A maps to any element in B.
  • Each element a in A maps to at least one unique element in B. (correct)
  • Each element in A must map to multiple elements in B.
  • Every element in B must be mapped by one element in A.

What is the total number of functions that can be formed from set A with m elements to set B with n elements?

<p>m^n (B)</p> Signup and view all the answers

What is the range of the constant function f defined by f(x) = c for all x ∈ R?

<p>{c} (D)</p> Signup and view all the answers

What defines the identity function?

<p>f(x) = x for all x in its domain (C)</p> Signup and view all the answers

What is the range of the Signum function?

<p>{-1, 0, 1} (A)</p> Signup and view all the answers

Which of the following functions is defined as f(x) = |x|?

<p>Absolute value function (B)</p> Signup and view all the answers

How many constant functions can be defined from a set A with m elements to a set B with n elements?

<p>n (D)</p> Signup and view all the answers

Flashcards

Cartesian Product

The set of all possible ordered pairs formed by taking one element from each of two sets, denoted as A × B.

Number of Relations

The number of possible relations from a set A to a set B is 2 raised to the power of the product of the number of elements in A and B (2^(n(A) × n(B))).

Function Definition

A relation where every element in the domain maps to a unique element in the codomain.

Total Number of Functions

If set A has 'm' elements and set B has 'n' elements, there are n^m possible functions from A to B.

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Domain and Codomain

The domain of a function is the set of all possible input values. The codomain is the set of all possible output values. The range is the subset of the codomain containing all actual output values.

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Constant Function

A function where every input maps to the same constant output value.

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Identity Function

A function where every input maps to itself as the output.

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Absolute Value Function

A function that outputs the positive value of an input, regardless of its sign.

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Signum Function

A function that returns: 1 if input is positive, -1 if input is negative, and 0 if input is zero.

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Equality of Functions

Two functions are equal if they have the same domain and map the same output to each input value in their shared domain.

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Study Notes

Relations and Functions

  • A set of ordered pairs formed by elements from two non-empty sets is called the Cartesian product.
  • The number of relations from set A to set B, where A and B have 'm' and 'n' elements respectively, is 2m*n.
  • A relation is a function if for every element in the domain, there's a unique corresponding element in the range.
  • If a set A has 'm' elements and set B has 'n' elements, the total number of functions from A to B is nm.
  • The set A is the domain of a function, and the set B is the co-domain.
  • The range of a function consists of all outputs from the function.

Domain, Range, and Constant Function

  • The domain of a function includes all permissible input values (x).
  • The range of a function comprises the set of all possible output values (y).
  • A function defined as f(x) = c (where c is a constant) is a constant function.
    • Domain: All real numbers (ℝ)
    • Range: {c}

Identity Function

  • An identity function is where f(x) = x for all x.
  • Domain: All real numbers (ℝ)
  • Range: All real numbers (ℝ)

Absolute Value Function

  • An absolute value function is defined as f(x) = |x|.
    • f(x) = x if x > 0
    • f(x) = -x if x < 0
  • Domain: All real numbers (ℝ)
  • Range: Non-negative real numbers ([0, ∞))

Signum Function

  • The signum function is defined as f(x) = x/|x| if x ≠ 0, and f(x) = 0 if x = 0.
  • Commonly abbreviated as sgn x
  • Domain: All real numbers (ℝ)
  • Range: {-1, 0, 1}

Equal Functions

  • Two functions f and g are equal if:
    • Their domains are equal
    • For every x in the domain, f(x) = g(x)

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Description

This quiz covers fundamental concepts of relations and functions in mathematics. You'll explore Cartesian products, domains, ranges, and the characteristics of constant and identity functions. Test your understanding and grasp the essential principles of these mathematical topics.

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