Sets, relations & functions worksheet with answers

Free printable Sets, relations & functions worksheet: 6 problems from easy to exam level, with an answer key. Print it or save it as a PDF.

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1.Easy

The Venn diagram shows every element of the universal set U and of the sets A and B. Find |(A′ − B′) − (A ∩ B)|, the number of elements of (A′ − B′) − (A ∩ B).

A = {2, 5, 6, 8, 9, 12, 13, 19}B = {1, 5, 9, 13, 19}|U| = 10

|(A′ − B′) − (A ∩ B)| =

2.Easy

How many subsets of A = {1, 2, 3, 4, 5, 6, 7} contain neither 5 nor 6? Call the number N.

A = {1, 2, 3, 4, 5, 6, 7}

N =

3.Medium

R = {(1, 3), (2, 1), (3, 3), (4, 3)} is a relation on {1, 2, 3, 4}. What is the smallest number k of ordered pairs that must be added to R to make it transitive (the size of the transitive closure minus |R|)?

R = {(1, 3), (2, 1), (3, 3), (4, 3)}

k =

4.Medium

In a group of 40 students, 23 speak French (F), 17 speak German (G) and 19 speak Spanish (S). 13 speak French and German, 9 French and Spanish, 5 German and Spanish, and 3 speak all three. How many are in none of the three sets?

|U| = 40

|(F ∪ G ∪ S)′| =

5.Hard

How many bijections f: A → B are there if |A| = |B| = 4? Call the number N.

|A| = 4|B| = 4

N =

6.Hard

On X = {1, 2, …, 13} the relation R holds when x ≡ y (mod 3). How many ordered pairs does R contain? Call the number |R|.

X = {1, …, 13}x R y ⟺ 3 | (x − y)

|R| =

Answer key

1. |(A′ − B′) − (A ∩ B)| = 1
2. N = 32
3. k = 1
4. |(F ∪ G ∪ S)′| = 5
5. N = 24
6. |R| = 57

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