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🔬 Tutorial problems alpha

🔬 Tutorial problems alpha#

Note

This problems are designed to help you practice the concepts covered in the lectures. Not all problems may be covered in the tutorial, those left are for additional practice.

\(\alpha\).1#

Consider the sets \(A = \{2,3,4\}\), \(B = \{2,5,6\}\), \(C = \{5,6,2\}\), and \(D = \{6\}\).

(a) Determine whether each of the following statements are true, false, or ambiguous:

  1. \(4 \in C\),

  2. \(5 \in C\),

  3. \(A \subseteq B\),

  4. \(D \subseteq C\),

  5. \(B = C\),

  6. \(A = B\).

(b) Find each of the following sets:

  1. \(A \cap B\),

  2. \(A \cup B\),

  3. \(A \setminus B\),

  4. \(B \setminus A\),

  5. \(A \triangle B\),

  6. \((A \cup B) \setminus (A \cap B)\),

  7. \(A \cup B \cup C \cup D\),

  8. \(A \cap B \cap C\),

  9. \(A \cap B \cap C \cap D\).

[Sydsæter, Hammond, Strøm, and Carvajal, 2016] Exercises for Section 1.1, Question 1

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\(\alpha\).2#

Suppose that \(U\) is the set of all students at a particular university, \(F\) is the set of female students at that university, \(M\) is the set of mathematics students at that university, \(C\) is the set of students in the choir at that university, \(B\) is the set of biology students at that university, and \(T\) is the set of students at that university who play tennis.

(a) Describe the following sets using English prose (that is, in words):

  1. \(F \cap B \cap C\),

  2. \(M \cap F\),

  3. \((M \cap B) \setminus C\), and

  4. \(((M \cap B) \setminus C) \setminus T\).

(b) Write each of the following statements in (mathematical) set terminology.

  1. All biology students are mathematics students.

  2. There are female biology students in the university choir.

  3. No tennis player studies biology.

  4. Those female students who neither play tennis nor belong to the university choir all study biology.

[Sydsæter, Hammond, Strøm, and Carvajal, 2016] Exercises for Section 1.1, Question 2.

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\(\alpha\).3#

A survey revealed that 50 people liked coffee and 40 liked tea. Both of these figures include 35 who liked both coffee and tea. Finally, ten did not like either coffee or tea. How many people in all responded to the survey? Explain your reasoning.

Hint

Drawing a Venn diagram for this question can be very helpful.

[Sydsæter, Hammond, Strøm, and Carvajal, 2016] Exercises for Section 1.1, Question 3.

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\(\alpha\).4#

Make a complete list of all the different subsets of the set \(\{a,b,c\}\). How many such subsets are there if the empty set and the set itself are included? Repeat this question for the set \(\{a, b, c, d\}\).

[Sydsæter, Hammond, Strøm, and Carvajal, 2016] Exercises for Section 1.1, Question 4

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\(\alpha\).5#

One thousand people took part in a survey to reveal which newspaper (\(A\), \(B\), or \(C\)) that had read on a (common) specified day. The responses showed that 420 had read \(A\), 316 had read \(B\), and 160 had read \(C\). These figures include 116 who had read both \(A\) and \(B\), 100 who had read both \(A\) and \(C\), and 30 who had read both \(B\) and \(C\). Finally, all of these figures include 16 who had read all three papers.

  1. How many respondents to the survey had read \(A\) but not \(B\)?

  2. How many respondents to the survey had read \(C\), but neither \(A\) nor \(B\)?

  3. How many respondents to the survey had read neither \(A\) nor \(B\) nor \(C\)?

[Sydsæter, Hammond, Strøm, and Carvajal, 2016] Exercises for Section 1.1, Question 8

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