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Introduction to sets

Below is a clear, complete, exam-oriented explanation of Introduction to Sets as used in Discrete Structures.

📘 Introduction to Sets

A set is a well-defined collection of distinct objects. These objects are called elements or members of the set.

  • “Well-defined” means that we can clearly decide whether an object belongs to the set or not.
  • “Distinct” means no element is repeated.

Examples:

  • ( A = {1, 2, 3, 4} )
  • ( B = {\text{apple}, \text{banana}, \text{mango}} )
  • ( C = {x \mid x \text{ is an even number less than 20}} )

📌 Ways to Describe a Set

1. Roster / Tabular Form

List elements inside curly brackets.

  • ( S = {2, 4, 6, 8} )

2. Set-builder Form

Specifies a rule/properties of elements.

  • ( S = {x \mid x \text{ is an even number less than 10}} )

📌 Notations

  • ( x \in A ): x is an element of A
  • ( x \notin A ): x is not an element of A
  • ( |A| ): cardinality of A (number of elements)
  • ( A = B ): Both sets have exactly the same elements

📌 Types of Sets

1. Empty Set (Null Set)

A set with no elements.
( \phi ) or ( {} )

2. Singleton Set

A set with exactly one element.
Example: ( {5} )

3. Finite Set

A set with a countable number of elements.
Example: ( {1,2,3,4} )

4. Infinite Set

A set with infinitely many elements.
Example: ( {1,2,3,4,\dots} )

5. Equal Sets

Two sets are equal when they have the same elements.
Example:
( A = {1,2,3}, B = {3,2,1} \Rightarrow A = B )

6. Equivalent Sets

Two sets with the same number of elements.
Example:
( {a,b,c} ) and ( {1,2,3} ) are equivalent.

7. Subset

A is a subset of B if every element of A is in B.
Notation: ( A \subseteq B )

Example:
( {1,2} \subseteq {1,2,3} )

8. Proper Subset

( A \subset B ) and ( A \neq B )

9. Universal Set (U)

The set that contains all elements under consideration.
Example:
If studying alphabets of English:
( U = {A, B, C, \ldots, Z} )

📌 Operations on Sets

1. Union

( A \cup B ): All elements in A or B or both.

2. Intersection

( A \cap B ): Elements common to both A and B.

3. Difference

( A – B ): Elements in A but not in B.

4. Complement

( A’ ) or ( A^c ): All elements in universal set U but not in A.

5. Cartesian Product

( A \times B = {(a,b) \mid a \in A, b \in B} )

📌 Venn Diagrams

Graphical representation of sets showing relationships like union, intersection, etc.

✔️ Key Properties of Sets

  • Idempotent Laws:
    ( A \cup A = A ), ( A \cap A = A )
  • 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) )
  • Distributive Laws:
    ( A \cap (B \cup C) = (A \cap B) \cup (A \cap C) )