Podcast
Questions and Answers
The number of symmetric relations defined on the set X which are not reflexive is:
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:
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?
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:
If A = {1, 2, 3, 4, 5, 6, 7}, then the relation R = {(x, y) ∈ A × A : x + y = 7} is:
If f : N → Z and f(x) = $x^2$, then:
If f : N → Z and f(x) = $x^2$, then:
Flashcards are hidden until you start studying
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.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.