# Set Theory and venn diagrams

2. Describe (in words) a Universal Set U containing a great many items, such as "All humans" or "All countries." Then describe (in words) two different sets that belong to U. Describe (in words) the complement of each of these two sets. Do any members of the Universal Set U lie outside both your first set and its complement?

3. Using the same two sets you described in problem 2, describe (in words) the new set formed by joining the two sets together. Is the new set called the union of the original sets, the intersection of the original sets, or neither? Explain.

1. Answer "Yes" or "No" for each question

- If A B and B C, can you conclude that A C? Can you conclude that A C?

- If A B and B C, can you conclude that A C? Can you conclude that A C?

- If A B and B C, can you conclude that A C? Can you conclude that A C?

2. Write down all possible subsets of {x, y, z}

3. Without writing them down, what are the number of subsets of the set A = {a, b, c, d, e}? Of set B = {a, b, c, d, e, f, g, h, i}?

4. Given U = {All letters of the alphabet}, A = {c, d, e, f}, and B = {e, f, g, h, k}. List the elements of set

(a) A U B

(b) A ∩ B

(c) A′ ∩ B′

(d) A′ U B′

(e) A U B′

(f) (A U B′) ∩ B

(g) (A U B) ∩ (A U B′)

5. Write the following in roster form: Set N is the set of natural numbers between two and nine.

6. State whether set A and B are equal, equivalent, both, or neither.

A = {9, 8, 7} B = {8, 9, 10}

7. Express the following in set-builder notation: M = {3, 4, 5, 6, 7}

8. A drug company is considering manufacturing a new product that has two different flavors, orange and cherry. They surveyed 150 people. The results are as follows:

75 liked cherry flavor

94 liked orange flavor

22 liked both flavors.

Construct a Venn diagram and answer the following:

a) How many liked only orange flavor?

b) How many liked only cherry flavor?

c) How many liked either one or the other or both?

d) How many liked neither?

9. In a survey of 75 resorts, it was reported that:

45 provided refrigerators in the guest rooms

37 provided laundry services

44 provided business centers

23 provided refrigerators in the guest rooms and laundry services

29 provided refrigerators in the guest rooms and business centers

21 provided laundry services and business centers

12 provided all three features

Construct a Venn diagram and use it to answer the following questions:

(a) How many of the resorts provided only refrigerators in the guest rooms?

(b) How many of the resorts provided exactly one of the features?

(c) How many of the resorts provided at least one of the features?

(d) How many of the resorts provided exactly two of the features?

(e) How many of the resorts provided none of the features?

Here are some symbols you might need in your answers: Є U ∩ ≤ Є ′ COPY AND PASTE!

10. Write the following in roster form:

Set N is the set of natural numbers between four and eleven.

11. Express the following in set builder notation:

Z = {9, 10, 11, 12, 13, 14, 15, 16, 17}

12. For sets A and B, determine whether A = B, A is a subset of B, or B is a subset of A

A = {x| x Є N and 13 < x < 20}

B = {14, 15, 16, 17, 18}

13. Given the diagram below, find a) P ∩ Q and b) P U Q

14. Given U = {21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31}, A = {21, 23, 25, 27, 29}, and

B = {22, 25, 28, 29}. Find A′ U B′.

15. Given U = {l, m, n, o, p, q, r, s, t, u, v, w}, A = {l, o, p, q, s, t}, B = {n, o, r, s, v, w},

and C = {l, m, n, q, r, t}, find (A′ U C′) ∩ B ′.

16. Set K contains 50 elements, set J contains 66 elements, and 14 elements are common to both sets. Find n (K U J).

17. Given A = {1, 2, 3, 4}, B = {3, 4, 5, 6,}, and C = {4, 6, 7}. Evaluate each set

a) A ∩ B

b) A ∩ C

c) A U C

d) B U C

e) (A U B) ∩ C

f) A U (B U C)

g) (A ∩ B) ∩ C

h) (A ∩ B) U C

18. In a survey of 100 consumers, 33 indicated that they were going to buy a new car, 18 said they were going to buy a new refrigerator, and 34 said they were going to buy a new washer. Of these, 7 were going to buy both a car and a refrigerator, 15 were going to buy a car and a washer, and 9 were going to buy a washer and a refrigerator. Three consumers indicated that they were going to buy all three items.

Construct a Venn diagram, label your diagram clearly.

Use your diagram to answer the following questions:

(a) How many were going to buy only a car?

(b) How many were going to buy only a washer?

(c) How many were going to buy only a refrigerator

(d) How many were going to buy a car and a washer but not a refrigerator?

(e) How many were going to buy none of these items?

19. A survey of a group of people produced the following results: there were 35 people with brown eyes and 24 people with blonde hair. If 14 people had both brown eyes and blonde hair and 27 people had neither, how many people were interviewed?

20. A survey of residents in a small town showed the following:

25 ate beef 13 ate both beef and fish 9 ate all types

28 ate fish 14 ate both fish and poultry 7 ate none of them

30 ate poultry 15 ate both beef and poultry

Draw a Venn diagram. Label your diagram clearly.

Use your diagram to answer the following questions:

(a) How many ate fish, but not poultry or beef?

(b) How many ate poultry and fish, but not beef?

(c) How many ate only beef?

(d) How many ate nothing or only poultry?

https://brainmass.com/math/discrete-math/set-theory-and-venn-diagrams-250824

#### Solution Summary

Venn diagrams and set theory are discussed.

Set Theory/Venn Diagrams

A recent survey of a group of students found that many participate in baseball, football and soccer. The Venn diagram below shows the results of the survey.

a) How many students participated in the survey?

b) How many of these students play both soccer and baseball?

c) How many play only 1 sport?

Given A = {a, b, c, d, e}

B = { a, b, d, f, g}

C = {b, c, e, g, h}

D = {d, e, f, g, h}

Find:

a) A B

b) (A D) B

c) A (B D)2