Injectivity and Surjectivity of Functions Quiz

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Which of the following relations is reflexive, symmetric, and transitive?

Relation R in the set Z of all integers defined as R = {(x, y) : x – y is an integer} (correct)

Relation R in the set A = {1, 2, 3, 4, 5, 6} as R = {(x, y) : y is divisible by x}

Relation R in the set N of natural numbers defined as R = {(x, y) : y = x + 5 and x < 4}

Relation R in the set A = {1, 2, 3,..., 13, 14} defined as R = {(x, y) : 3x – y = 0}

Which of the following relations is neither reflexive, symmetric, nor transitive?

Relation R in R defined by R = {(a, b) : a ≤ b3}

Relation R defined in the set {1, 2, 3, 4, 5, 6} as R = {(a, b) : b = a + 1}

Relation R in R defined as R = {(a, b) : a ≤ b}

Relation R in the set R of real numbers, defined as R = {(a, b) : a ≤ b2} (correct)

Which of the following relations is reflexive and transitive, but not symmetric?

Relation R in the set N of natural numbers defined as R = {(x, y) : y = x + 5 and x < 4}

Relation R in the set A = {1, 2, 3,..., 13, 14} defined as R = {(x, y) : 3x – y = 0}

Relation R in the set A = {1, 2, 3, 4, 5, 6} as R = {(x, y) : y is divisible by x}

Relation R in R defined as R = {(a, b) : a ≤ b} (correct)

Which of the following relations is symmetric but neither reflexive nor transitive?

Relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} Signup and view all the answers

Which of the following relations is reflexive, but neither symmetric nor transitive?

Relation R in R defined as R = {(a, b) : a ≤ b} Signup and view all the answers

What type of relation is the relation R in the set R of real numbers defined by R = {(a, b) : a ≤ b3}?

Neither reflexive, symmetric, nor transitive Signup and view all the answers

Which of the following functions is bijective?

None of the above Signup and view all the answers

For the function f: R → R given by f(x) = [x], where [x] denotes the greatest integer less than or equal to x, which of the following properties is true?

f is neither one-one nor onto Signup and view all the answers

For the Modulus Function f: R → R given by f(x) = |x|, where |x| is x if x is positive or 0, and -x if x is negative, which of the following statements is correct?

f is neither one-one nor onto Signup and view all the answers

For the Signum Function f: R → R given by f(x) = 1 if x > 0, f(x) = 0 if x = 0, and f(x) = -1 if x < 0, which of the following properties holds?

f is neither one-one nor onto Signup and view all the answers

Let A = {1, 2, 3}, B = {4, 5, 6, 7}, and f = {(1, 4), (2, 5), (3, 6)} be a function from A to B. Which of the following properties does f possess?

f is one-one (injective) Signup and view all the answers

Consider the function f: R → R defined by f(x) = 3 - 4x. Which of the following properties does f possess?

f is one-one (injective) Signup and view all the answers

Which of the following best describes an equivalence relation?

A relation where all elements are related to each other Signup and view all the answers

In the context of the given relation R, why are all elements of the subset {1, 3, 5, 7} related to each other?

Because they are all odd Signup and view all the answers

Why can no element from the subset {1, 3, 5, 7} be related to any element from the subset {2, 4, 6}?

Because odd numbers cannot be related to even numbers Signup and view all the answers

If a relation is reflexive, it means that:

Every element is related to itself Signup and view all the answers

Symmetry in an equivalence relation implies that:

If (a, b) is in the relation, then (b, a) must also be in the relation Signup and view all the answers

Transitivity in an equivalence relation means that:

(a, b) and (b, c) being in the relation implies (a, c) is also in the relation Signup and view all the answers

Injectivity and Surjectivity of Functions Quiz

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Questions and Answers

Which of the following relations is reflexive, symmetric, and transitive?

Which of the following relations is neither reflexive, symmetric, nor transitive?

Which of the following relations is reflexive and transitive, but not symmetric?

Which of the following relations is symmetric but neither reflexive nor transitive?

Which of the following relations is reflexive, but neither symmetric nor transitive?

What type of relation is the relation R in the set R of real numbers defined by R = {(a, b) : a ≤ b3}?

Which of the following functions is bijective?

For the function f: R → R given by f(x) = [x], where [x] denotes the greatest integer less than or equal to x, which of the following properties is true?

For the Modulus Function f: R → R given by f(x) = |x|, where |x| is x if x is positive or 0, and -x if x is negative, which of the following statements is correct?

For the Signum Function f: R → R given by f(x) = 1 if x > 0, f(x) = 0 if x = 0, and f(x) = -1 if x < 0, which of the following properties holds?

Let A = {1, 2, 3}, B = {4, 5, 6, 7}, and f = {(1, 4), (2, 5), (3, 6)} be a function from A to B. Which of the following properties does f possess?

Consider the function f: R → R defined by f(x) = 3 - 4x. Which of the following properties does f possess?

Which of the following best describes an equivalence relation?

In the context of the given relation R, why are all elements of the subset {1, 3, 5, 7} related to each other?

Why can no element from the subset {1, 3, 5, 7} be related to any element from the subset {2, 4, 6}?

If a relation is reflexive, it means that:

Symmetry in an equivalence relation implies that:

Transitivity in an equivalence relation means that:

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