Sets, Their Types and Representations

A set is a well-defined collection of distinct objects called elements or members. Sets are denoted by capital letters such as \(A, B, C\), and elements are listed within curly brackets, e.g., \(A = \{1, 2, 3\}\).

Types of Sets

• Empty or Null Set: Contains no elements, denoted by \(\emptyset\) or \(\{\}\).
• Singleton Set: Contains exactly one element, e.g., \(\{0\}\).
• Finite Set: Has a limited number of elements, e.g., days of the week.
• Infinite Set: Has unlimited elements, e.g., all integers.

Subset

Set \(A\) is a subset of set \(B\) (written \(A \subseteq B\)) if every element of \(A\) is also in \(B\).

Cardinal Number

The cardinal number \(n(A)\) of a set \(A\) is the number of elements in \(A\). For example, if \(A = \{a,b,c,d,e\}\), then \(n(A) = 5\).

Set Representation

• Roster Form: List all elements explicitly, e.g., \(\{1, 3, 5, 7, 9\}\).
• Set Builder Form: Define elements by a property, e.g., \(A = \{x \mid x \text{ is a prime number less than } 10\}\).

Equal and Equivalent Sets

• Equal Sets: Sets with exactly the same members, e.g., \(\{1, 2, 3\} = \{3, 2, 1\}\).
• Equivalent Sets: Sets with the same number of elements but not necessarily the same members, e.g., \(A = \{1, 2, 3\}\) and \(B = \{a, b, c\}\) are equivalent since \(n(A) = n(B) = 3\).

Worked Example

Given \(A = \{2, 4, 6\}\) and \(B = \{4, 6, 8, 10\}\), determine if \(A\) is a subset of \(B\).

Solution:

Check if every element of \(A\) is in \(B\):

• 2 is not in \(B\).

Therefore, \(A \not\subseteq B\).

Practice Set

Level 1 – Easy

• List the elements of the set \(C = \{x \mid x \text{ is an even number less than } 10\}\).
• Is \(\emptyset\) a subset of every set?

Level 2 – Moderate

• Find the cardinal number of \(D = \{1, 3, 5, 7, 9\}\).
• Are the sets \(E = \{1, 2, 3\}\) and \(F = \{3, 2, 1\}\) equal?

Level 3 – Challenging

• Prove that the set \(G = \{x \in \mathbb{N} \mid x^2 < 20\}\) is finite and find its cardinal number.
• Determine if \(H = \{x \in \mathbb{Z} \mid -3 \leq x \leq 3\}\) is a subset of \(\mathbb{Z}\).

Answer Key

• Level 1:
• \(C = \{2, 4, 6, 8\}\)
• Yes, \(\emptyset\) is a subset of every set.
• Level 2:
• \(n(D) = 5\)
• Yes, \(E = F\) because they have the same elements.
• Level 3:
• \(G = \{1, 2, 3, 4\}\), finite with \(n(G) = 4\).
• Yes, \(H \subseteq \mathbb{Z}\) since all elements of \(H\) are integers.

Quick Reference

Term	Notation	Definition
Empty Set	\(\emptyset\)	No elements
Subset	\(A \subseteq B\)	All elements of \(A\) are in \(B\)
Cardinal Number	\(n(A)\)	Number of elements in \(A\)
Equal Sets	\(A = B\)	Same elements
Equivalent Sets	\(A \leftrightarrow B\)	Same cardinality

Glossary

• Element: An object in a set.
• Set: Collection of distinct elements.
• Subset: A set contained within another set.
• Cardinal Number: Number of elements in a set.
• Empty Set: Set with no elements.
• Singleton Set: Set with one element.
• Finite Set: Set with countable elements.
• Infinite Set: Set with uncountable elements.

Venn Diagrams and Operations on Sets

A Venn diagram visually represents sets and their relationships using circles within a rectangle representing the universal set \(U\).

Universal Set

The universal set \(U\) contains all elements under consideration.

Set Operations

Union

The union of sets \(A\) and \(B\), denoted \(A \cup B\), is the set of elements in \(A\) or \(B\) or both:

\[ A \cup B = \{x \mid x \in A \text{ or } x \in B\} \]

Intersection

The intersection of sets \(A\) and \(B\), denoted \(A \cap B\), is the set of elements common to both:

\[ A \cap B = \{x \mid x \in A \text{ and } x \in B\} \]

Disjoint Sets

Sets \(A\) and \(B\) are disjoint if \(A \cap B = \emptyset\), meaning they have no elements in common.

Difference of Sets

The difference \(A - B\) is the set of elements in \(A\) but not in \(B\):

\[ A - B = \{x \mid x \in A \text{ and } x \notin B\} \]

Similarly, \(B - A = \{x \mid x \in B \text{ and } x \notin A\}\).

Symmetric Difference

The symmetric difference \(A \Delta B\) is the set of elements in either \(A\) or \(B\) but not in both:

\[ A \Delta B = (A - B) \cup (B - A) \]

Complement

The complement of \(A\) with respect to \(U\), denoted \(A'\) or \(A^c\), is:

\[ A' = U - A = \{x \mid x \in U \text{ and } x \notin A\} \]

Properties and Laws

• Complement Laws:
• \(A \cup A' = U\)
• \(A \cap A' = \emptyset\)
• De Morgan's Laws:
• \((A \cup B)' = A' \cap B'\)
• \((A \cap B)' = A' \cup B'\)
• Double Complement: \((A')' = A\)
• Empty and Universal Set Complements:
• \(\emptyset' = U\)
• \(U' = \emptyset\)
• Commutative Laws:
• \(A \cup B = B \cup A\)
• \(A \cap B = B \cap A\)
• Associative Laws:
• \((A \cup B) \cup C = A \cup (B \cup C)\)
• \((A \cap B) \cap C = A \cap (B \cap C)\)
• Distributive Laws:
• \(A \cap (B \cup C) = (A \cap B) \cup (A \cap C)\)
• \(A \cup (B \cap C) = (A \cup B) \cap (A \cup C)\)

Worked Example

Given \(U = \{1,2,3,4,5,6,7,8\}\), \(A = \{2,4,6,8\}\), and \(B = \{1,2,3,4\}\), find \(A \cup B\), \(A \cap B\), \(A - B\), and \(A'\).

Solution:

• \(A \cup B = \{1,2,3,4,6,8\}\)
• \(A \cap B = \{2,4\}\)
• \(A - B = \{6,8\}\)
• \(A' = U - A = \{1,3,5,7\}\)

Practice Set

Level 1 – Easy

• Draw a Venn diagram for sets \(A = \{1,2,3\}\) and \(B = \{3,4,5\}\).
• Find \(A \cup B\) and \(A \cap B\) for the above sets.

Level 2 – Moderate

• Prove that \(A \cup \emptyset = A\) for any set \(A\).
• Find the complement of \(B = \{2,4,6\}\) in \(U = \{1,2,3,4,5,6,7\}\).

Level 3 – Challenging

• Verify De Morgan's law: \((A \cup B)' = A' \cap B'\) using sets \(A = \{1,2,3\}\), \(B = \{3,4,5\}\), and \(U = \{1,2,3,4,5,6\}\).
• Show that if \(A \subseteq B\), then \(A \cap B = A\) and \(A \cup B = B\).

Answer Key

• Level 1:
• \(A \cup B = \{1,2,3,4,5\}\)
• \(A \cap B = \{3\}\)
• Level 2:
• \(A \cup \emptyset = A\) because union with empty set adds no elements.
• Complement of \(B\) in \(U\) is \(\{1,3,5,7\}\).
• Level 3:
• \((A \cup B)' = \{6\}\), \(A' = \{4,5,6\}\), \(B' = \{1,2,6\}\), and \(A' \cap B' = \{6\}\). Hence, verified.
• If \(A \subseteq B\), then \(A \cap B = A\) and \(A \cup B = B\) by definition.

Quick Reference

Operation	Notation	Definition
Union	\(A \cup B\)	Elements in \(A\) or \(B\) or both
Intersection	\(A \cap B\)	Elements common to \(A\) and \(B\)
Difference	\(A - B\)	Elements in \(A\) not in \(B\)
Symmetric Difference	\(A \Delta B\)	Elements in \(A\) or \(B\) but not both
Complement	\(A'\)	Elements in \(U\) not in \(A\)

Glossary

• Universal Set: The set containing all elements under consideration.
• Union: Combination of elements from two sets.
• Intersection: Common elements of two sets.
• Difference: Elements in one set but not the other.
• Complement: Elements not in the given set but in the universal set.
• Disjoint Sets: Sets with no common elements.
• Symmetric Difference: Elements in either set but not in both.