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Free Practice Quiz Question List

03 Relations and Functions Online Quiz Questions

Use this free practice quiz with 20 questions to review 03 Relations and Functions, test your knowledge, and prepare for your next test or exam.

20 questions
01
Choose all
1 point

Which two properties are required for a relation to be an equivalence relation, in addition to transitivity? Select all correct choices.

  1. A

    Reflexive

  2. B

    Antisymmetric

  3. C

    Symmetric

  4. D

    Surjective

02
Choose all
1 point

Which statements describe properties of a function? Select all correct choices.

  1. A

    Every input has at least two outputs.

  2. B

    Distinct inputs have distinct outputs.

  3. C

    The domain and codomain must be identical sets.

  4. D

    Every codomain element is the output of at least one input.

03
Written response
1 point

What property does the function f:Z→Zf:\mathbb Z\to\mathbb Z defined by f(n)=2n+1f(n)=2n+1 have because distinct integers produce distinct outputs?

04
Fill in the blank
1 point

Complete the definition: A relation is a partial order when it is , , and .

05
Fill in the blank
1 point

Congruence modulo a positive integer is an whose equivalence classes form a of the integers.

06
Choose one
1 point

Let A={1,2}A=\{1,2\} and B={a,b}B=\{a,b\}. Which relation from AA to BB is a function?

  1. A

    {(1,a),(2,b)}\{(1,a),(2,b)\}

  2. B

    {(1,a),(1,b),(2,a)}\{(1,a),(1,b),(2,a)\}

  3. C

    {(1,a)}\{(1,a)\}

  4. D

    {(1,a),(2,a),(2,b)}\{(1,a),(2,a),(2,b)\}

07
Choose one
1 point

For the relation R={(1,a),(2,a),(2,b)}R=\{(1,a),(2,a),(2,b)\}, which pair gives its domain and range?

  1. A

    Domain {1,2,3}\{1,2,3\}; range {a,b}\{a,b\}

  2. B

    Domain {1,2}\{1,2\}; range {a,b}\{a,b\}

  3. C

    Domain {a,b}\{a,b\}; range {1,2}\{1,2\}

  4. D

    Domain {1,2}\{1,2\}; range {a}\{a\}

08
Choose one
1 point

Which function is not injective?

  1. A

    f(n)=2n+1f(n)=2n+1, f:Z→Zf:\mathbb Z\to\mathbb Z

  2. B

    f(n)=3n−2f(n)=3n-2, f:Z→Zf:\mathbb Z\to\mathbb Z

  3. C

    g(n)=n2g(n)=n^2, g:Z→Zg:\mathbb Z\to\mathbb Z

  4. D

    h(n)=n+5h(n)=n+5, h:Z→Zh:\mathbb Z\to\mathbb Z

09
Choose one
1 point

Which function is not surjective onto its stated codomain?

  1. A

    f(x)=x3f(x)=x^3, with f:R→Rf:\mathbb R\to\mathbb R

  2. B

    g(x)=x2g(x)=x^2, with g:R→Rg:\mathbb R\to\mathbb R

  3. C

    h(x)=2x+1h(x)=2x+1, with h:R→Rh:\mathbb R\to\mathbb R

  4. D

    k(x)=x−7k(x)=x-7, with k:R→Rk:\mathbb R\to\mathbb R

10
Choose one
1 point

Which integer belongs to the equivalence class [1][1] modulo 33?

  1. A

    44

  2. B

    55

  3. C

    66

  4. D

    88

11
True or false
1 point

True or false: Distinct equivalence classes of an equivalence relation are disjoint.

  1. A

    True

  2. B

    False

12
Written response
1 point

Let f:R→Rf:\mathbb R\to\mathbb R be defined by f(x)=3x−4f(x)=3x-4. Enter only the right-hand side of the formula for f−1(x)f^{-1}(x), not the complete equation.

13
Written response
1 point

If f:A→Bf:A\to B is a bijection and AA has 77 elements, how many elements does BB have?

14
Open ended
1 point

Explain why g:R→Rg:\mathbb R\to\mathbb R, defined by g(x)=x2g(x)=x^2, is not injective. Then explain how restricting its domain to [0,∞)[0,\infty) changes its injectivity.

15
True or false
1 point

True or false: A relation can be both symmetric and antisymmetric.

  1. A

    True

  2. B

    False

16
Choose one
1 point

Let f:{1,2,3}→{a,b}f:\{1,2,3\}\to\{a,b\} be f={(1,a),(2,a),(3,b)}f=\{(1,a),(2,a),(3,b)\}. Why is the inverse relation not a function from {a,b}\{a,b\} to {1,2,3}\{1,2,3\}?

  1. A

    It is a function because every element of AA occurs.

  2. B

    It is a function because the original function has three ordered pairs.

  3. C

    It is not a function because aa is paired with both 11 and 22.

  4. D

    It is not a function because bb is not an element of AA.

17
Choose one
1 point

Let f:{1,2,3}→{a,b,c}f:\{1,2,3\}\to\{a,b,c\} be defined by f(1)=af(1)=a, f(2)=af(2)=a, and f(3)=bf(3)=b. Which statement correctly describes its image and codomain?

  1. A

    The image is {a,b}\{a,b\}, while the codomain is {a,b,c}\{a,b,c\}.

  2. B

    The image is {a,b,c}\{a,b,c\}, while the codomain is {a,b}\{a,b\}.

  3. C

    The image and codomain are both {a,b}\{a,b\}.

  4. D

    This rule is not a function because two inputs have the same output.

18
True or false
1 point

True or false: For every set SS, the relation ⊆\subseteq on P(S)\mathcal P(S) is a total order.

  1. A

    True

  2. B

    False

19
Written response
1 point

What standard term describes a function that is both injective and surjective?

20
Choose one
1 point

Which set is a valid relation from A={1,2}A=\{1,2\} to B={a,b}B=\{a,b\}?

  1. A

    {(1,a),(3,c)}\{(1,a),(3,c)\}, for A={1,2}A=\{1,2\} and B={a,b}B=\{a,b\}

  2. B

    {(1,a),(2,c)}\{(1,a),(2,c)\}, for A={1,2}A=\{1,2\} and B={a,b}B=\{a,b\}

  3. C

    {(1,a),(2,b)}\{(1,a),(2,b)\}, for A={1,2}A=\{1,2\} and B={a,b}B=\{a,b\}

  4. D

    {(a,1),(b,2)}\{(a,1),(b,2)\}, for A={1,2}A=\{1,2\} and B={a,b}B=\{a,b\}