समुच्चय (Sets)

A. What is a Set?

A Set is a well-defined collection of distinct objects. "Well-defined" means there is an unambiguous rule that determines whether any given object belongs to the collection or not:

• Valid Set: "The vowels in the English alphabet" $\to \{a, e, i, o, u\}$ (Universal agreement).
• NOT a Set: "The collection of honest people in India" (Vague and subjective; no universal definition of "honest").

B. Two Forms of Representation:

1. Roster / Tabular Form: Elements are listed explicitly separated by commas inside curly brackets $\{ \}$: $$V = \{a, e, i, o, u\}$$ *(Order does not matter, and elements are never repeated: $\{1, 2, 2, 3\} = \{1, 2, 3\}$)*.
2. Set-Builder / Rule Form: A common characterizing property $P(x)$ shared by all elements is stated: $$A = \{x : x \in \mathbb{N} \text{ and } 3 < x \le 8\} = \{4, 5, 6, 7, 8\}$$

A. Types of Sets:

• Empty / Null Set ($\emptyset$ or $\{ \}$): Contains zero elements ($n(\emptyset) = 0$). *(Note: $\{0\}$ is NOT empty; it is a singleton containing the number 0!)*.
• Singleton Set: Contains exactly one element (e.g., $\{7\}$).
• Cardinal Number $n(A)$: The count of distinct elements in a finite set $A$.
• Equal Sets ($A = B$): Exactly the same elements.
• Equivalent Sets ($A \approx B$): Same number of elements: $n(A) = n(B)$.

B. Subsets & Power Set:

• Subset ($A \subseteq B$): Every element of $A$ is also an element of $B$.
• The empty set $\emptyset$ is a subset of every set: $\emptyset \subseteq A$.
• Every set is a subset of itself: $A \subseteq A$.
• Number of Subsets: A set containing $n$ elements has strictly $2^n$ subsets and $2^n - 1$ proper subsets.
• Power Set $P(A)$: The set of all subsets of $A$. If $A = \{1, 2\}$, then: $$P(A) = \{\emptyset, \{1\}, \{2\}, \{1, 2\}\} \quad [n(P(A)) = 2^2 = 4]$$

1. Union ($A \cup B$): Set of elements belonging to $A$, or $B$, or both: $$A \cup B = \{x : x \in A \lor x \in B\}$$
2. Intersection ($A \cap B$): Set of elements common to BOTH $A$ and $B$: $$A \cap B = \{x : x \in A \land x \in B\}$$ *(If $A \cap B = \emptyset$, the sets are called Disjoint Sets)*.
3. Difference ($A - B$): Elements belonging to $A$ but NOT belonging to $B$: $$A - B = \{x : x \in A \land x \notin B\}$$
4. Complement of a Set ($A'$ or $A^c$): Elements of the Universal Set $U$ that do not belong to $A$: $$\mathbf{A' = U - A = \{x \in U : x \notin A\}}$$

De Morgan's Laws:

In a Venn Diagram, the Universal set $U$ is represented by a large outer rectangle, and subsets are drawn as intersecting or disjoint circles within it.

The Fundamental Cardinality Formula for Two Sets:

Rearrangement: $n(A \cap B) = n(A) + n(B) - n(A \cup B)$.

• Elements in only $A$: $n(A - B) = n(A) - n(A \cap B)$.
• Elements in only $B$: $n(B - A) = n(B) - n(A \cap B)$.
• Total elements: $n(A \cup B) = n(A - B) + n(B - A) + n(A \cap B)$.