Set Notation (Number of Elements, Complement, Universal Set, Union, Intersection)

Introduction

Key Concepts

1. Sets and Set Notation

A **set** is a collection of distinct objects, considered as an object in its own right. These objects are called **elements** or **members** of the set. Set notation provides a standardized way to describe and manipulate sets.

Sets are usually denoted by uppercase letters, and their elements are listed within curly braces. For example, the set of natural numbers less than 5 is written as:

If a set has no elements, it is called an **empty set** and is denoted by:

2. Number of Elements (Cardinality)

The **number of elements** in a set is referred to as its **cardinality**. It is denoted by vertical bars surrounding the set, for example, |A| represents the cardinality of set A.

**Example:** If $$ A = \{2, 4, 6, 8, 10\} $$ then $$ |A| = 5 $$ because there are five elements in set A.

3. Universal Set

The **universal set**, often denoted by U, is the set that contains all the objects or elements under consideration for a particular discussion or problem. All other sets are subsets of the universal set.

**Example:** If we are discussing even numbers less than 10, the universal set could be: $$ U = \{1, 2, 3, 4, 5, 6, 7, 8, 9\} $$ and a subset A of U could be: $$ A = \{2, 4, 6, 8\} $$

4. Complement of a Set

The **complement** of a set A, denoted by A', consists of all elements in the universal set U that are not in A.

Mathematically, it is expressed as: $$ A' = \{x \in U \mid x \notin A\} $$

**Example:** Given the universal set $$ U = \{1, 2, 3, 4, 5, 6, 7, 8, 9\} $$ and set $$ A = \{2, 4, 6, 8\} $$ the complement of A is: $$ A' = \{1, 3, 5, 7, 9\} $$

5. Union of Sets

The **union** of two sets A and B, denoted by $$ A \cup B $$ , is the set containing all elements that are in A, in B, or in both.

**Formula:** $$ A \cup B = \{x \mid x \in A \text{ or } x \in B\} $$

**Example:** If $$ A = \{1, 2, 3\} $$ and $$ B = \{3, 4, 5\} $$ then $$ A \cup B = \{1, 2, 3, 4, 5\} $$

6. Intersection of Sets

The **intersection** of two sets A and B, denoted by $$ A \cap B $$ , is the set containing all elements that are both in A and in B.

**Formula:** $$ A \cap B = \{x \mid x \in A \text{ and } x \in B\} $$

**Example:** If $$ A = \{1, 2, 3\} $$ and $$ B = \{3, 4, 5\} $$ then $$ A \cap B = \{3\} $$

7. Subsets and Proper Subsets

A set A is a **subset** of set B, denoted by $$ A \subseteq B $$ , if every element of A is also an element of B.

If A is a subset of B but not equal to B, A is called a **proper subset**, denoted by $$ A \subset B $$ .

**Example:** If $$ A = \{1, 2\} $$ and $$ B = \{1, 2, 3\} $$ then $$ A \subset B $$ .

8. Venn Diagrams

**Venn diagrams** are visual representations of sets and their relationships. They help in understanding operations like union, intersection, and complement.

In a Venn diagram:

• Each set is represented by a circle.
• The universal set is usually depicted by a rectangle enclosing all sets.
• The overlapping areas represent the intersection of sets.
• Non-overlapping areas represent elements unique to each set.

**Example:** Consider sets A and B: $$ A = \{1, 2, 3\} $$ $$ B = \{3, 4, 5\} $$ The union $$ A \cup B = \{1, 2, 3, 4, 5\} $$ is represented by the entire area covered by both circles, while the intersection $$ A \cap B = \{3\} $$ is the overlapping region.

9. De Morgan's Laws

**De Morgan's Laws** provide a relationship between union and intersection through complements.

The laws are stated as:

• The complement of the union of two sets is equal to the intersection of their complements: $$ (A \cup B)' = A' \cap B' $$
• The complement of the intersection of two sets is equal to the union of their complements: $$ (A \cap B)' = A' \cup B' $$

**Example:** Given sets A = {1, 2, 3} and B = {3, 4, 5} with universal set U = {1, 2, 3, 4, 5, 6}, then: $$ (A \cup B)' = \{6\} $$ and $$ A' \cap B' = \{6\} $$ showing that $$ (A \cup B)' = A' \cap B' $$

10. Power Set

The **power set** of a set A, denoted by $$ \mathcal{P}(A) $$ , is the set of all possible subsets of A, including the empty set and A itself.

**Formula:** If $$ |A| = n $$ , then $$ |\mathcal{P}(A)| = 2^n $$ .

**Example:** For $$ A = \{1, 2\} $$ , the power set is: $$ \mathcal{P}(A) = \{\emptyset, \{1\}, \{2\}, \{1, 2\}\} $$

11. Disjoint Sets

Two sets A and B are **disjoint** if they have no elements in common, that is, $$ A \cap B = \emptyset $$ .

**Example:** If $$ A = \{1, 2\} $$ and $$ B = \{3, 4\} $$ then A and B are disjoint sets.

12. Cartesian Product

The **Cartesian product** of two sets A and B, denoted by $$ A \times B $$ , is the set of all ordered pairs (a, b) where $$ a \in A $$ and $$ b \in B $$ .

**Formula:** $$ A \times B = \{(a, b) \mid a \in A \text{ and } b \in B\} $$

**Example:** If $$ A = \{1, 2\} $$ and $$ B = \{x, y\} $$ , then: $$ A \times B = \{(1, x), (1, y), (2, x), (2, y)\} $$

Advanced Concepts

1. Set Difference

The **difference** between two sets A and B, denoted by $$ A - B $$ or $$ A \setminus B $$ , is the set of elements that are in A but not in B.

**Formula:** $$ A - B = \{x \mid x \in A \text{ and } x \notin B\} $$

**Example:** If $$ A = \{1, 2, 3, 4\} $$ and $$ B = \{3, 4, 5\} $$ , then: $$ A - B = \{1, 2\} $$

2. Symmetric Difference

The **symmetric difference** of two sets A and B, denoted by $$ A \triangle B $$ , is the set of elements which are in either of the sets A or B but not in their intersection.

**Formula:** $$ A \triangle B = (A - B) \cup (B - A) $$

**Example:** If $$ A = \{1, 2, 3\} $$ and $$ B = \{3, 4, 5\} $$ , then: $$ A \triangle B = \{1, 2, 4, 5\} $$

3. Indexed Sets

**Indexed sets** are sets where the elements are indexed by another set, usually a number set. This concept is useful in defining sequences and functions.

**Example:** Consider an indexed set where $$ A_i = i^2 $$ for $$ i = 1, 2, 3, 4 $$ , then: $$ A = \{1, 4, 9, 16\} $$

4. Infinite Sets

An **infinite set** is a set that has no end; it contains an endless number of elements. Examples include the set of natural numbers, integers, real numbers, etc.

**Example:** The set of natural numbers: $$ \mathbb{N} = \{1, 2, 3, 4, 5, \ldots\} $$ is an infinite set.

5. Finite vs. Infinite Sets

A **finite set** has a definite number of elements, whereas an **infinite set** does not. Distinguishing between the two is essential in different areas of mathematics, including calculus and discrete mathematics.

**Example:** - Finite set: $$ A = \{a, b, c\} $$ - Infinite set: $$ \mathbb{R} \text{ (set of all real numbers)} $$

6. Countable and Uncountable Sets

A **countable set** is either finite or has the same cardinality as the set of natural numbers, meaning its elements can be listed in a sequence. An **uncountable set** has strictly greater cardinality, meaning its elements cannot be listed in a sequence.

**Example:** - Countable set: $$ \mathbb{Z} \text{ (set of all integers)} $$ - Uncountable set: $$ \mathbb{R} \text{ (set of all real numbers)} $$

7. Applications of Set Theory

Set theory is not only fundamental in pure mathematics but also has applications across various disciplines:

• Computer Science: Used in database theory, formal languages, and algorithms.
• Logic: Forms the basis for mathematical logic and reasoning.
• Probability and Statistics: Sets are used to define events and sample spaces.
• Engineering: Applied in systems design and analysis.

8. Advanced Problem-Solving Techniques

Understanding set operations allows for solving complex problems involving relationships between different groups or categories.

**Example:** In a survey, 100 students were asked about their preferences for sports. If 60 students like football (F), 50 like basketball (B), and 30 like both, determine how many students like only football, only basketball, or neither.

**Solution:** - Number of students who like only football: $$ |F| - |F \cap B| = 60 - 30 = 30 $$ - Number of students who like only basketball: $$ |B| - |F \cap B| = 50 - 30 = 20 $$ - Number of students who like neither: $$ U - (|F| + |B| - |F \cap B|) = 100 - (60 + 50 - 30) = 100 - 80 = 20 $$

9. Proof Techniques in Set Theory

Proofs in set theory often involve demonstrating inclusion, equality, or properties of sets using logical reasoning and previously established theorems.

**Example Proof:** Prove that $$ A \cup (B \cap C) = (A \cup B) \cap (A \cup C) $$

**Proof:** To prove that two sets are equal, show that each is a subset of the other.

• First, show $$ A \cup (B \cap C) \subseteq (A \cup B) \cap (A \cup C) $$ .
• Take any element $$ x \in A \cup (B \cap C) $$ .
• If $$ x \in A $$ , then $$ x \in A \cup B $$ and $$ x \in A \cup C $$ , so $$ x \in (A \cup B) \cap (A \cup C) $$ .
• If $$ x \in B \cap C $$ , then $$ x \in B $$ and $$ x \in C $$ , so $$ x \in A \cup B $$ and $$ x \in A \cup C $$ , hence $$ x \in (A \cup B) \cap (A \cup C) $$ .
• Second, show $$ (A \cup B) \cap (A \cup C) \subseteq A \cup (B \cap C) $$ .
• Take any element $$ x \in (A \cup B) \cap (A \cup C) $$ .
• This means $$ x \in A \cup B $$ and $$ x \in A \cup C $$ .
• If $$ x \in A $$ , then $$ x \in A \cup (B \cap C) $$ .
• If $$ x \notin A $$ , then $$ x \in B $$ and $$ x \in C $$ , so $$ x \in B \cap C $$ , hence $$ x \in A \cup (B \cap C) $$ .

Since each set is a subset of the other, $$ A \cup (B \cap C) = (A \cup B) \cap (A \cup C) $$ .

10. Infinite Set Operations

Operations on infinite sets require careful handling, especially when dealing with concepts like cardinality and different types of infinities.

**Example:** Consider the universal set $$ U = \mathbb{N} \text{ (set of natural numbers)} $$ and subsets $$ A = \{2, 4, 6, \ldots\} \text{ (even numbers)} $$ , $$ B = \{3, 6, 9, \ldots\} \text{ (multiples of 3)} $$ .

The union $$ A \cup B = \{2, 3, 4, 6, 8, 9, \ldots\} $$ and the intersection $$ A \cap B = \{6, 12, 18, \ldots\} $$ .

Despite being infinite, these operations follow the same principles as finite sets, illustrating the consistency of set theory across different magnitudes.

Comparison Table

Summary and Key Takeaways