Relations and Functions in Math

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Questions and Answers

The number of symmetric relations defined on the set X which are not reflexive is:

  • 860
  • 850
  • 960 (correct)
  • 930

Let A be the set of all subsets of X. Then the relation R is:

  • Symmetric and reflexive only
  • Reflexive only
  • Symmetric only (correct)
  • Symmetric and transitive only

In the given relation on X, which statement is true?

  • Only statement (I) is correct. (correct)
  • Only statement (II) is correct.
  • Both statements (I) and (II) are correct.
  • Neither statement (I) nor (II) is correct.

If A = {1, 2, 3, 4, 5, 6, 7}, then the relation R = {(x, y) ∈ A × A : x + y = 7} is:

<p>An equivalence relation (A)</p> Signup and view all the answers

If f : N → Z and f(x) = $x^2$, then:

<p>f(x) is bijective (B)</p> Signup and view all the answers

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Study Notes

Relation and Function

  • A relation R on a set A is symmetric if (a, b) ∈ R implies (b, a) ∈ R.
  • A relation R on a set A is reflexive if (a, a) ∈ R for each a ∈ A.
  • A relation R on a set A is transitive if (a, b) ∈ R and (b, c) ∈ R imply (a, c) ∈ R.

Examples of Relations

  • Let A = {1, 2, 3, 4, 5, 6, 7}. The relation R = {(x, y) ∈ A × A : x + y = 7} is symmetric but neither reflexive nor transitive.
  • Let A = {1, 2, 3, 4, 5, 6, 7}. The relation R = {(x, y) ∈ A × A : x + y = 7} is not an equivalence relation.

Functions

  • A function f : A → B is bijective if it is both injective and surjective.
  • A function f : A → B is injective if for every b ∈ B, there exists at most one a ∈ A such that f(a) = b.
  • A function f : A → B is surjective if for every b ∈ B, there exists at least one a ∈ A such that f(a) = b.

Examples of Functions

  • Let f : N → Z be defined by f(x) = x^2 + 3x - 2. Then f(x) is not injective but surjective.
  • Let f : [2, 3] → R be defined by f(x) = x^3 + 3x - 2. Then the range of f(x) is contained in the interval [1, 12].
  • Let f : R → R be defined by f(x) = x^2 - 6x - 14. Then f^(-1)(2) equals {-2, 8}.

Inverse of Functions

  • The inverse of a function f is denoted by f^(-1).
  • The inverse of a function f is a function g such that g(f(x)) = x for every x in the domain of f.

Periodic Functions

  • A function f is periodic if there exists a positive real number T such that f(x + T) = f(x) for every x in the domain of f.
  • The period of a function f is the smallest positive real number T such that f(x + T) = f(x) for every x in the domain of f.

Examples of Periodic Functions

  • The function f(x) = sin(x) is periodic with period 2π.
  • The function f(x) = cos(x) is periodic with period 2π.
  • The function f(x) = tan(x) is periodic with period π.

Graphs of Functions

  • The graph of a function f is the set of all points (x, y) such that y = f(x).
  • The graph of a function f can be used to determine the domain and range of f.

Examples of Graphs of Functions

  • The graph of the function f(x) = x^2 is a parabola.
  • The graph of the function f(x) = sin(x) is a sine wave.
  • The graph of the function f(x) = cos(x) is a cosine wave.

Equivalence Relations

  • An equivalence relation on a set A is a relation R that is reflexive, symmetric, and transitive.
  • An equivalence relation on a set A partitions A into equivalence classes.

Examples of Equivalence Relations

  • The relation R = {(x, y) ∈ A × A : x - y is an integer} is an equivalence relation on the set of real numbers.
  • The relation R = {(x, y) ∈ A × A : x^2 = y^2} is an equivalence relation on the set of real numbers.

Reflexive, Symmetric, and Transitive Relations

  • A relation R is reflexive if (a, a) ∈ R for every a ∈ A.
  • A relation R is symmetric if (a, b) ∈ R implies (b, a) ∈ R.
  • A relation R is transitive if (a, b) ∈ R and (b, c) ∈ R imply (a, c) ∈ R.

Examples of Reflexive, Symmetric, and Transitive Relations

  • The relation R = {(x, y) ∈ A × A : x + y = 7} is symmetric but neither reflexive nor transitive.
  • The relation R = {(x, y) ∈ A × A : x - y is an integer} is reflexive, symmetric, and transitive.

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