Sets

1. Definition of a Set

A set is a well-defined collection of distinct objects, called elements. A set is denoted by capital letters like $A$, $B$, $C$, and its elements are written inside braces { }.

Example: $A = \{1, 2, 3\}$

2. Elements of a Set

The individual objects in a set are called its elements. If an element $a$ belongs to set $A$, we write $a \in A$. If not, $a \notin A$.

Example: If $A = \{2, 4, 6\}$, then $4 \in A$, but $5 \notin A$.

3. Order of Sets

Sets are unordered. Thus, $\{1,2,3\}$ and $\{3,1,2\}$ represent the same set.

4. Representation of Sets

• Statement Form: The set of even numbers less than 10.
• Roster Form: $A = \{2, 4, 6, 8\}$
• Set Builder Form: $A = \{x : x \text{ is an even number less than } 10\}$

5. Types of Sets

• Empty Set: $\varnothing$ or $\{\}$
• Finite and Infinite Sets: $\{1,2,3\}$ is finite; $\mathbb{N}=\{1,2,3,\dots\}$ is infinite.
• Equal Sets: If they have exactly the same elements.
• Subset: $A \subseteq B$ if every element of $A$ is also in $B$.
• Proper Subset: $A \subset B$ means $A$ is a subset but not equal to $B$.
• Universal Set: The set containing all objects under consideration, usually denoted by $U$.

6. Operations on Sets

• Union: $A \cup B = \{x : x \in A \text{ or } x \in B\}$
• Intersection: $A \cap B = \{x : x \in A \text{ and } x \in B\}$
• Difference: $A - B = \{x : x \in A \text{ and } x \notin B\}$
• Complement: $A' = \{x : x \in U \text{ and } x \notin A\}$

7. Union of Sets

8. Intersection of Sets

9. Cartesian Product of Sets

10. Complement of Sets

The complement of a set $A$ is denoted by $A'$ or $A^c$ and includes all elements of the universal set $U$ that are not in $A$.

11. Difference of Sets

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

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

12. Power Set

The set of all subsets of $A$ is called its power set, denoted $\mathcal{P}(A)$.

If $A = \{1,2\}$, then $\mathcal{P}(A) = \{\varnothing, \{1\}, \{2\}, \{1,2\}\}$.

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

13. Number of Elements in Sets

The cardinality of a finite set $A$ is the number of elements in $A$, written as $|A|$.

For finite sets $A$ and $B$:

• $|A \cup B| = |A| + |B| - |A \cap B|$
• If $A$ and $B$ are disjoint, $|A \cup B| = |A| + |B|$