Chapter 1: Sets

1.1 Introduction to Sets

A set is a well-defined collection of objects. These objects are called the elements or members of the set. Sets are usually denoted by capital letters: \( A, B, C, X, Y, Z \), etc. The elements of a set are represented by small letters: \( a, b, c, x, y, z \), etc.

If \( a \) is an element of set \( A \), we write \( a \in A \) (read as “\( a \) belongs to \( A \)”). If \( b \) is not an element of \( A \), we write \( b \notin A \).

Representations of Sets

1. Roster or Tabular Form: All elements of the set are listed, separated by commas, and enclosed within braces \( \{ \} \). Example: The set of all vowels in the English alphabet is \( V = \{a, e, i, o, u\} \).
2. Set-Builder Form: All elements possess a single common property which is not possessed by any element outside the set. Example: \( V = \{x : x \text{ is a vowel in English alphabet}\} \).

1.2 Types of Sets

• Empty Set: A set which does not contain any element. Denoted by \( \emptyset \) or \( \{\} \).
• Finite and Infinite Sets: A set which is empty or consists of a definite number of elements is finite, otherwise it is infinite.
• Equal Sets: Two sets \( A \) and \( B \) are equal if they have exactly the same elements. We write \( A = B \).
• Subsets: A set \( A \) is a subset of \( B \) if every element of \( A \) is also an element of \( B \). We write \( A \subseteq B \).
• Universal Set: A broad set that contains all elements under consideration, denoted by \( U \).

Subsets of Real Numbers

Important numerical sets and their standard notations:

• \( \mathbb{N} \): Set of natural numbers \( \{1, 2, 3, \ldots\} \)
• \( \mathbb{Z} \): Set of integers \( \{\ldots, -2, -1, 0, 1, 2, \ldots\} \)
• \( \mathbb{Q} \): Set of rational numbers
• \( \mathbb{R} \): Set of real numbers

Intervals as Subsets of \( \mathbb{R} \)

• Open interval: \( (a, b) = \{x : a < x < b\} \)
• Closed interval: \( [a, b] = \{x : a \le x \le b\} \)
• Semi-open intervals: \( [a, b) = \{x : a \le x < b\} \) and \( (a, b] = \{x : a < x \le b\} \)

1.3 Venn Diagrams

Most relationships between sets can be visualized using diagrams, known as Venn Diagrams. The Universal Set \( U \) is represented by a rectangle, and its subsets are represented by circles.

1.4 Operations on Sets

1. Union of Sets: The union of \( A \) and \( B \) is the set of all elements which are either in \( A \), or in \( B \), or in both. \[ A \cup B = \{x : x \in A \text{ or } x \in B\} \]
2. Intersection of Sets: The intersection is the set of elements common to both \( A \) and \( B \). \[ A \cap B = \{x : x \in A \text{ and } x \in B\} \]
3. Difference of Sets: The difference \( A - B \) is the set of elements which belong to \( A \) but not to \( B \). \[ A - B = \{x : x \in A \text{ and } x \notin B\} \]
4. Complement of a Set: Given a universal set \( U \) and a subset \( A \), the complement of \( A \) is the set of all elements of \( U \) which are not elements of \( A \). \[ A’ = \{x : x \in U \text{ and } x \notin A\} = U - A \]

Properties of Complement

• Complement Laws: \( A \cup A’ = U \) and \( A \cap A’ = \emptyset \)
• De Morgan’s Laws:
  • \( (A \cup B)’ = A’ \cap B’ \)
  • \( (A \cap B)’ = A’ \cup B’ \)

Competency-Based Questions

Question 1:

A marketing firm conducted a survey of 1000 people to find out their preferences for three different brands of coffee: A, B, and C. It was found that 450 preferred brand A, 350 preferred brand B, and 300 preferred brand C. Furthermore, 150 preferred both A and B, 100 preferred both B and C, and 120 preferred both A and C. If 80 people liked all three brands, how many people do not prefer any of the three brands? Identify this via set operations.

Answer 1:

Let \( U \) be the set of surveyed people. Thus, \( n(U) = 1000 \).

Let \( A, B, C \) represent the sets of people preferring brands A, B, and C respectively.

We are given:

\( n(A) = 450 \), \( n(B) = 350 \), \( n(C) = 300 \)

\( n(A \cap B) = 150 \)

\( n(B \cap C) = 100 \)

\( n(A \cap C) = 120 \)

\( n(A \cap B \cap C) = 80 \)

We need to find the number of people who do NOT prefer any brand, which is \( n( (A \cup B \cup C)’ ) \).

First, calculate the union of all three sets:

\[ n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(B \cap C) - n(A \cap C) + n(A \cap B \cap C) \]

\[ n(A \cup B \cup C) = 450 + 350 + 300 - 150 - 100 - 120 + 80 \]

\[ n(A \cup B \cup C) = 1100 - 370 + 80 = 810 \]

The number of people who do not prefer any brand:

\[ n( (A \cup B \cup C)’ ) = n(U) - n(A \cup B \cup C) = 1000 - 810 = 190 \]

Conclusion: 190 people evaluated in the survey did not prefer any of the three brands.

Question 2:

Let \( U = \{1, 2, 3, 4, 5, 6, 7, 8, 9\} \), \( A = \{2, 4, 6, 8\} \), and \( B = \{2, 3, 5, 7\} \).

Verify De Morgan’s Law: \( (A \cup B)’ = A’ \cap B’ \).

Answer 2:

Step 1: Find \( A \cup B \).

\( A \cup B = \{2, 3, 4, 5, 6, 7, 8\} \)

Step 2: Find \( (A \cup B)’ \).

\( (A \cup B)’ = U - (A \cup B) = \{1, 9\} \)

Step 3: Find \( A’ \) and \( B’ \).

\( A’ = U - A = \{1, 3, 5, 7, 9\} \)

\( B’ = U - B = \{1, 4, 6, 8, 9\} \)

Step 4: Find \( A’ \cap B’ \).

\( A’ \cap B’ = \{1, 9\} \)

Conclusion: Since \( (A \cup B)’ = \{1, 9\} \) and \( A’ \cap B’ = \{1, 9\} \), De Morgan’s Law \( (A \cup B)’ = A’ \cap B’ \) is verified.