# 1.1 Sets

## Detailed Theory: Sets

### **1. Basic Concepts and Definitions of Sets**

#### **1.1 What is a Set?**

A set is a **well-defined collection** of **distinct objects**. The objects in a set are called its **elements** or **members**.

**Key Characteristics:**

* **Well-defined:** It must be clear whether a particular object belongs to the set or not
* **Distinct:** No element is repeated (each element appears only once)
* **Order doesn't matter:** $$\ {1, 2, 3} = {3, 1, 2} = {2, 3, 1}$$

#### **1.2 Notation and Symbols**

* Sets are denoted by capital letters: $$A, B, C, \ldots$$
* Elements are denoted by lowercase letters: $$a, b, c, \ldots$$
* **Belongs to:** $$a \in A$$ means "a is an element of A"
* **Does not belong to:** $$a \notin A$$ means "a is not an element of A"
* **Such that:** $$|$$ or $$:$$ (colon) means "such that"

**Example:** $$A = {1, 2, 3, 4, 5}$$ Here, $$3 \in A$$ but $$7 \notin A$$

#### **1.3 Methods of Describing Sets**

**a) Roster/Tabular Method**

All elements are listed within braces {}, separated by commas.

**Examples:**

1. Set of vowels in English alphabet: $$V = {a, e, i, o, u}$$
2. Set of first five natural numbers: $$N\_5 = {1, 2, 3, 4, 5}$$
3. Set of even numbers between 1 and 10: $$E = {2, 4, 6, 8, 10}$$

**b) Set-Builder Method**

Specifies a property that characterizes all elements of the set.

**General Form:** $${x : P(x)}$$ or $${x \mid P(x)}$$ Read as: "The set of all x such that P(x) is true"

**Examples:**

1. $${x : x \text{ is a natural number less than 6}}$$ This means: $${1, 2, 3, 4, 5}$$
2. $${x \in \mathbb{Z} : -3 < x < 4}$$ This means: $${-2, -1, 0, 1, 2, 3}$$
3. $${x : x = 2n, n \in \mathbb{N}}$$ This means the set of all even natural numbers: $${2, 4, 6, 8, \ldots}$$

#### **1.4 Types of Sets**

**a) Empty/Null Set**

A set with **no elements**. Denoted by $$\emptyset$$ or $${}$$.

**Important:** $${\emptyset}$$ is NOT an empty set. It's a set containing one element (which is the empty set).

**Examples:**

* Set of natural numbers less than 1: $$\emptyset$$
* $${x : x^2 = -1, x \in \mathbb{R}} = \emptyset$$

**b) Singleton Set**

A set with **exactly one element**.

**Examples:**

* $$A = {5}$$
* $$B = {\emptyset}$$ (This is a singleton set containing the empty set)
* Set of solutions of $$x + 2 = 5$$: $${3}$$

**c) Finite Set**

A set with finite number of elements. We can count all elements.

**Examples:**

* $$A = {a, b, c, d}$$ has 4 elements
* Set of days in a week: $${Sunday, Monday, \ldots, Saturday}$$ has 7 elements
* $$B = {x : x \text{ is a prime number less than 10}} = {2, 3, 5, 7}$$ has 4 elements

**d) Infinite Set**

A set with infinite number of elements. We cannot count all elements.

**Examples:**

* Set of all natural numbers: $$\mathbb{N} = {1, 2, 3, \ldots}$$
* Set of all integers: $$\mathbb{Z} = {\ldots, -2, -1, 0, 1, 2, \ldots}$$
* Set of all real numbers: $$\mathbb{R}$$

**e) Equal Sets**

Two sets A and B are equal if they have **exactly the same elements**.

**Notation:** $$A = B$$

**Criteria:** $$A = B$$ if and only if every element of A is in B and every element of B is in A.

**Examples:**

1. $${1, 2, 3} = {3, 1, 2} = {2, 1, 3}$$
2. $${x : x^2 - 3x + 2 = 0} = {1, 2}$$
3. $${a, b, c} \neq {a, b, d}$$

**f) Equivalent Sets**

Two sets are equivalent if they have the **same number of elements** (same cardinality).

**Notation:** $$A \sim B$$

**Important:** Equal sets are always equivalent, but equivalent sets may not be equal.

**Examples:**

* $$A = {1, 2, 3}$$ and $$B = {a, b, c}$$ are equivalent (both have 3 elements)
* $${1, 2, 3} \sim {4, 5, 6}$$ (both have cardinality 3)

***

### **2. Subsets and Power Sets**

#### **2.1 Subset**

Set A is a **subset** of set B if every element of A is also an element of B.

**Notation:** $$A \subseteq B$$ Read as: "A is a subset of B" or "A is contained in B"

**Formal Definition:** $$A \subseteq B$$ if and only if $$x \in A \Rightarrow x \in B$$ for all x

**Examples:**

1. $${1, 2} \subseteq {1, 2, 3, 4}$$
2. $$\mathbb{N} \subseteq \mathbb{Z} \subseteq \mathbb{Q} \subseteq \mathbb{R} \subseteq \mathbb{C}$$
3. Every set is a subset of itself: $$A \subseteq A$$

#### **2.2 Proper Subset**

A is a **proper subset** of B if $$A \subseteq B$$ but $$A \neq B$$.

**Notation:** $$A \subset B$$ or $$A \subsetneq B$$

**Examples:**

1. $${1, 2} \subset {1, 2, 3}$$
2. $$\emptyset \subset {1, 2, 3}$$ (Empty set is a proper subset of every non-empty set)
3. $${a, b} \subset {a, b, c}$$

#### **2.3 Important Facts about Subsets**

1. **Empty set is subset of every set:** $$\emptyset \subseteq A$$ for any set A
2. **Reflexive property:** $$A \subseteq A$$ for any set A
3. **Transitive property:** If $$A \subseteq B$$ and $$B \subseteq C$$, then $$A \subseteq C$$
4. **Antisymmetric property:** If $$A \subseteq B$$ and $$B \subseteq A$$, then $$A = B$$

#### **2.4 Number of Subsets**

If a set A has n elements, then:

* Total number of subsets of A = $$2^n$$
* Number of proper subsets of A = $$2^n - 1$$

**Proof:** Each element has 2 choices: either be in a subset or not be in it. So for n elements, total possibilities = $$2 \times 2 \times \cdots \times 2$$ (n times) = $$2^n$$.

**Example:** For $$A = {a, b, c}$$ (n=3):

* Total subsets = $$2^3 = 8$$
* Proper subsets = $$2^3 - 1 = 7$$

#### **2.5 Power Set**

The set of **all subsets** of a given set A is called the power set of A.

**Notation:** $$P(A)$$ or $$2^A$$ or $$\mathcal{P}(A)$$

**Properties:**

1. $$\emptyset \in P(A)$$ and $$A \in P(A)$$
2. If A has n elements, then $$P(A)$$ has $$2^n$$ elements
3. $$A \subseteq B \Rightarrow P(A) \subseteq P(B)$$

**Example:** For $$A = {1, 2}$$: $$P(A) = {\emptyset, {1}, {2}, {1, 2}}$$

Here, n(A) = 2, and n(P(A)) = $$2^2 = 4$$

***

### **3. Universal Set and Complement**

#### **3.1 Universal Set**

The set that contains **all objects** under consideration in a particular context.

**Notation:** Usually denoted by U or $$\xi$$

**Properties:**

1. Every set under consideration is a subset of U
2. The choice of universal set depends on the context

**Examples:**

* In number theory: $$U = \mathbb{N}$$ or $$\mathbb{Z}$$
* In plane geometry: $$U = \text{set of all points in the plane}$$
* For a survey of students: $$U = \text{set of all students in the school}$$

#### **3.2 Complement of a Set**

The complement of a set A is the set of all elements in the universal set U that are **not** in A.

**Notation:** $$A'$$ or $$A^c$$ or $$\overline{A}$$ or $$U - A$$

**Formal Definition:** $$A' = {x : x \in U \text{ and } x \notin A}$$

**Venn Diagram Representation:** The region outside circle A but inside rectangle U.

**Examples:** Let $$U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}$$ and $$A = {2, 4, 6, 8, 10}$$ Then $$A' = {1, 3, 5, 7, 9}$$

#### **3.3 Properties of Complement**

1. **Complement Laws:**
   * $$A \cup A' = U$$
   * $$A \cap A' = \emptyset$$
2. **Double Complement Law:** $$(A')' = A$$
3. **Universal and Empty Set Complements:**
   * $$U' = \emptyset$$
   * $$\emptyset' = U$$
4. **De Morgan's Laws:**
   * $$(A \cup B)' = A' \cap B'$$
   * $$(A \cap B)' = A' \cup B'$$

**Proof of De Morgan's Law:** $$(A \cup B)' = A' \cap B'$$ Let $$x \in (A \cup B)'$$ $$\Rightarrow x \notin (A \cup B)$$ $$\Rightarrow x \notin A \text{ and } x \notin B$$ $$\Rightarrow x \in A' \text{ and } x \in B'$$ $$\Rightarrow x \in (A' \cap B')$$ Thus, $$(A \cup B)' \subseteq A' \cap B'$$

Conversely, let $$x \in A' \cap B'$$ $$\Rightarrow x \in A' \text{ and } x \in B'$$ $$\Rightarrow x \notin A \text{ and } x \notin B$$ $$\Rightarrow x \notin (A \cup B)$$ $$\Rightarrow x \in (A \cup B)'$$ Thus, $$A' \cap B' \subseteq (A \cup B)'$$

Therefore, $$(A \cup B)' = A' \cap B'$$

***

### **4. Operations on Sets**

#### **4.1 Union of Sets**

The union of sets A and B is the set of all elements that are in A, in B, or in both.

**Notation:** $$A \cup B$$

**Formal Definition:** $$A \cup B = {x : x \in A \text{ or } x \in B}$$

**Venn Diagram:** The entire shaded region of both circles.

**Examples:**

1. $${1, 2, 3} \cup {3, 4, 5} = {1, 2, 3, 4, 5}$$
2. $${a, b} \cup {c, d} = {a, b, c, d}$$
3. $${x : x > 0} \cup {x : x < 5} = \mathbb{R} - {0}$$ (all real numbers except 0)

#### **4.2 Intersection of Sets**

The intersection of sets A and B is the set of all elements common to both A and B.

**Notation:** $$A \cap B$$

**Formal Definition:** $$A \cap B = {x : x \in A \text{ and } x \in B}$$

**Venn Diagram:** The overlapping region of the two circles.

**Examples:**

1. $${1, 2, 3} \cap {3, 4, 5} = {3}$$
2. $${a, b, c} \cap {c, d, e} = {c}$$
3. $${x : x > 2} \cap {x : x < 5} = {x : 2 < x < 5}$$

**Disjoint Sets**

Two sets are disjoint if their intersection is empty.

**Definition:** A and B are disjoint if $$A \cap B = \emptyset$$

**Examples:**

* $${1, 2}$$ and $${3, 4}$$ are disjoint
* Set of even numbers and set of odd numbers are disjoint

#### **4.3 Difference of Sets**

The difference of sets A and B is the set of elements that are in A but not in B.

**Notation:** $$A - B$$ or $$A \setminus B$$

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

**Important:** $$A - B \neq B - A$$ in general

**Examples:**

1. $${1, 2, 3, 4} - {3, 4, 5} = {1, 2}$$
2. $${3, 4, 5} - {1, 2, 3, 4} = {5}$$
3. $${a, b, c} - {b, c, d} = {a}$$

**Properties:**

* $$A - B = A \cap B'$$
* $$A - A = \emptyset$$
* $$A - \emptyset = A$$
* $$\emptyset - A = \emptyset$$
* $$A - U = \emptyset$$

#### **4.4 Symmetric Difference**

The symmetric difference of sets A and B is the set of elements that are in either A or B but not in both.

**Notation:** $$A \triangle B$$ or $$A \ominus B$$

**Formal Definitions:**

1. $$A \triangle B = (A \cup B) - (A \cap B)$$
2. $$A \triangle B = (A - B) \cup (B - A)$$

**Venn Diagram:** The non-overlapping parts of both circles.

**Examples:**

1. $${1, 2, 3} \triangle {3, 4, 5} = {1, 2, 4, 5}$$
2. $${a, b, c} \triangle {b, c, d} = {a, d}$$

**Properties:**

1. **Commutative:** $$A \triangle B = B \triangle A$$
2. **Associative:** $$(A \triangle B) \triangle C = A \triangle (B \triangle C)$$
3. $$A \triangle A = \emptyset$$
4. $$A \triangle \emptyset = A$$
5. $$A \triangle U = A'$$

***

### **5. Laws of Set Algebra**

#### **5.1 Commutative Laws**

1. $$A \cup B = B \cup A$$
2. $$A \cap B = B \cap A$$

**Proof of** $$A \cup B = B \cup A$$**:** Let $$x \in A \cup B$$ $$\Rightarrow x \in A \text{ or } x \in B$$ $$\Rightarrow x \in B \text{ or } x \in A$$ $$\Rightarrow x \in B \cup A$$ Thus, $$A \cup B \subseteq B \cup A$$

Similarly, $$B \cup A \subseteq A \cup B$$ Therefore, $$A \cup B = B \cup A$$

#### **5.2 Associative Laws**

1. $$(A \cup B) \cup C = A \cup (B \cup C)$$
2. $$(A \cap B) \cap C = A \cap (B \cap C)$$

**Example Verification:** Let $$A = {1, 2}, B = {2, 3}, C = {3, 4}$$ LHS: $$(A \cup B) \cup C = ({1, 2, 3}) \cup {3, 4} = {1, 2, 3, 4}$$ RHS: $$A \cup (B \cup C) = {1, 2} \cup {2, 3, 4} = {1, 2, 3, 4}$$ LHS = RHS

#### **5.3 Distributive Laws**

1. $$A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$$
2. $$A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$$

**Proof of** $$A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$$**:** Let $$x \in A \cup (B \cap C)$$ Case 1: If $$x \in A$$, then $$x \in A \cup B$$ and $$x \in A \cup C$$ So $$x \in (A \cup B) \cap (A \cup C)$$

Case 2: If $$x \in B \cap C$$, then $$x \in B$$ and $$x \in C$$ So $$x \in A \cup B$$ and $$x \in A \cup C$$ Thus $$x \in (A \cup B) \cap (A \cup C)$$

Therefore, $$A \cup (B \cap C) \subseteq (A \cup B) \cap (A \cup C)$$

Conversely, let $$x \in (A \cup B) \cap (A \cup C)$$ Then $$x \in A \cup B$$ and $$x \in A \cup C$$

If $$x \in A$$, then $$x \in A \cup (B \cap C)$$ If $$x \notin A$$, then from $$x \in A \cup B$$, we get $$x \in B$$ And from $$x \in A \cup C$$, we get $$x \in C$$ So $$x \in B \cap C$$, hence $$x \in A \cup (B \cap C)$$

Therefore, $$(A \cup B) \cap (A \cup C) \subseteq A \cup (B \cap C)$$

Thus, $$A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$$

#### **5.4 Identity Laws**

1. $$A \cup \emptyset = A$$
2. $$A \cap U = A$$

#### **5.5 Complement Laws**

1. $$A \cup A' = U$$
2. $$A \cap A' = \emptyset$$

#### **5.6 Idempotent Laws**

1. $$A \cup A = A$$
2. $$A \cap A = A$$

#### **5.7 Absorption Laws**

1. $$A \cup (A \cap B) = A$$
2. $$A \cap (A \cup B) = A$$

#### **5.8 De Morgan's Laws (Already proved in section 3.3)**

1. $$(A \cup B)' = A' \cap B'$$
2. $$(A \cap B)' = A' \cup B'$$

#### **5.9 Involution Law**

$$(A')' = A$$

***

### **6. Cardinality of Sets**

#### **6.1 Cardinal Number**

The number of elements in a finite set A is called its cardinal number.

**Notation:** $$n(A)$$ or $$|A|$$ or $$\text{card}(A)$$

**Examples:**

1. For $$A = {a, b, c, d}$$, $$n(A) = 4$$
2. For $$B = {1, 3, 5, 7, 9}$$, $$n(B) = 5$$
3. $$n(\emptyset) = 0$$

#### **6.2 Inclusion-Exclusion Principle**

**For Two Sets:**

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

**Derivation:** When we add $$n(A)$$ and $$n(B)$$, elements in $$A \cap B$$ are counted twice. So we subtract $$n(A \cap B)$$ once to count them only once.

**Example:** Let $$A = {1, 2, 3, 4}, B = {3, 4, 5, 6}$$

* $$n(A) = 4, n(B) = 4$$
* $$A \cap B = {3, 4}, n(A \cap B) = 2$$
* $$A \cup B = {1, 2, 3, 4, 5, 6}, n(A \cup B) = 6$$
* Verify: $$6 = 4 + 4 - 2$$

**For Three Sets:**

$$n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(B \cap C) - n(C \cap A) + n(A \cap B \cap C)$$

**Example:** In a survey of 100 students:

* 60 like Math (M)
* 50 like Science (S)
* 40 like English (E)
* 30 like Math and Science
* 20 like Science and English
* 15 like Math and English
* 10 like all three

How many like at least one subject?

**Solution:** $$n(M \cup S \cup E) = 60 + 50 + 40 - 30 - 20 - 15 + 10 = 95$$

So 95 students like at least one subject.

#### **6.3 Cardinality Formulas**

1. $$n(A - B) = n(A) - n(A \cap B)$$
2. $$n(A \triangle B) = n(A) + n(B) - 2n(A \cap B)$$
3. $$n(A') = n(U) - n(A)$$
4. $$n(A \times B) = n(A) \cdot n(B)$$
5. $$n(P(A)) = 2^{n(A)}$$

***

### **7. Cartesian Product**

#### **7.1 Definition**

The Cartesian product of two sets A and B is the set of all ordered pairs (a, b) where a ∈ A and b ∈ B.

**Notation:** $$A \times B$$

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

**Important:**

* $$(a, b)$$ is an **ordered pair**, so $$(a, b) \neq (b, a)$$ unless a = b
* $$A \times B \neq B \times A$$ in general

#### **7.2 Examples**

1. Let $$A = {1, 2}, B = {a, b}$$ $$A \times B = {(1, a), (1, b), (2, a), (2, b)}$$ $$B \times A = {(a, 1), (a, 2), (b, 1), (b, 2)}$$ Clearly, $$A \times B \neq B \times A$$
2. $$\mathbb{R} \times \mathbb{R} = \mathbb{R}^2$$ (the Cartesian plane)
3. If $$A = \emptyset$$ or $$B = \emptyset$$, then $$A \times B = \emptyset$$

#### **7.3 Number of Elements**

If A has m elements and B has n elements, then $$A \times B$$ has $$m \times n$$ elements.

**Proof:** For each of the m elements in A, we can pair it with any of the n elements in B.

**Example:** For $$A = {1, 2, 3}$$ (m=3) and $$B = {a, b}$$ (n=2): $$n(A \times B) = 3 \times 2 = 6$$

#### **7.4 Properties of Cartesian Product**

1. **Non-commutative:** $$A \times B \neq B \times A$$ (unless A = B or one is empty)
2. **Non-associative:** $$(A \times B) \times C \neq A \times (B \times C)$$
3. **Distributive over union:** $$A \times (B \cup C) = (A \times B) \cup (A \times C)$$
4. **Distributive over intersection:** $$A \times (B \cap C) = (A \times B) \cap (A \times C)$$
5. **Distributive over difference:** $$A \times (B - C) = (A \times B) - (A \times C)$$

#### **7.5 Cartesian Product of Three or More Sets**

For sets A, B, C: $$A \times B \times C = {(a, b, c) : a \in A, b \in B, c \in C}$$

**Number of elements:** $$n(A \times B \times C) = n(A) \cdot n(B) \cdot n(C)$$

**Example:** $$A = {1, 2}, B = {a, b}, C = {x, y}$$ $$A \times B \times C = {(1, a, x), (1, a, y), (1, b, x), (1, b, y), (2, a, x), (2, a, y), (2, b, x), (2, b, y)}$$ Total elements = $$2 \times 2 \times 2 = 8$$

***

### **8. Solved Examples**

#### **Example 1:** Basic Set Operations

Let $$U = {1, 2, 3, 4, 5, 6, 7, 8}, A = {2, 4, 6, 8}, B = {3, 6, 7, 8}$$

Find:

1. $$A \cup B$$
2. $$A \cap B$$
3. $$A - B$$
4. $$B - A$$
5. $$A'$$
6. $$A \triangle B$$

**Solution:**

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

#### **Example 2:** Cardinality Problem

In a class of 50 students:

* 30 like Mathematics
* 25 like Physics
* 20 like Chemistry
* 15 like Mathematics and Physics
* 10 like Physics and Chemistry
* 8 like Mathematics and Chemistry
* 5 like all three subjects

Find:

1. How many like at least one subject?
2. How many like exactly one subject?
3. How many like none of the subjects?

**Solution:** Let M, P, C represent sets of students liking Mathematics, Physics, Chemistry respectively.

Given: $$n(U) = 50, n(M) = 30, n(P) = 25, n(C) = 20$$ $$n(M \cap P) = 15, n(P \cap C) = 10, n(M \cap C) = 8, n(M \cap P \cap C) = 5$$

1. Using inclusion-exclusion: $$n(M \cup P \cup C) = 30 + 25 + 20 - 15 - 10 - 8 + 5 = 47$$ So 47 students like at least one subject.
2. Students liking exactly one subject:
   * Only M: $$n(M) - n(M \cap P) - n(M \cap C) + n(M \cap P \cap C) = 30 - 15 - 8 + 5 = 12$$
   * Only P: $$n(P) - n(M \cap P) - n(P \cap C) + n(M \cap P \cap C) = 25 - 15 - 10 + 5 = 5$$
   * Only C: $$n(C) - n(M \cap C) - n(P \cap C) + n(M \cap P \cap C) = 20 - 8 - 10 + 5 = 7$$ Total = 12 + 5 + 7 = 24 students
3. Students liking none: $$n(U) - n(M \cup P \cup C) = 50 - 47 = 3$$

#### **Example 3:** Set Algebra Proof

Prove that $$A - (B \cup C) = (A - B) \cap (A - C)$$

**Proof:** Let $$x \in A - (B \cup C)$$ Then $$x \in A$$ and $$x \notin (B \cup C)$$ $$\Rightarrow x \in A$$ and $$x \notin B$$ and $$x \notin C$$ $$\Rightarrow x \in A$$ and $$x \notin B$$ and $$x \in A$$ and $$x \notin C$$ $$\Rightarrow x \in (A - B)$$ and $$x \in (A - C)$$ $$\Rightarrow x \in (A - B) \cap (A - C)$$ Thus, $$A - (B \cup C) \subseteq (A - B) \cap (A - C)$$

Conversely, let $$x \in (A - B) \cap (A - C)$$ Then $$x \in A - B$$ and $$x \in A - C$$ $$\Rightarrow x \in A$$ and $$x \notin B$$ and $$x \in A$$ and $$x \notin C$$ $$\Rightarrow x \in A$$ and $$x \notin B$$ and $$x \notin C$$ $$\Rightarrow x \in A$$ and $$x \notin (B \cup C)$$ $$\Rightarrow x \in A - (B \cup C)$$ Thus, $$(A - B) \cap (A - C) \subseteq A - (B \cup C)$$

Therefore, $$A - (B \cup C) = (A - B) \cap (A - C)$$

***

### **9. Practice Tips for Exams**

1. **Venn Diagrams:** Always draw Venn diagrams for visualization
2. **Set Notation:** Be precise with notation: $$\in, \subseteq, \subset, \cup, \cap, \setminus, \triangle$$
3. **Cardinality Formulas:** Memorize inclusion-exclusion principle
4. **Proof Techniques:** For proofs, use element method (show $$x \in LHS \Rightarrow x \in RHS$$ and vice versa)
5. **Special Cases:** Remember edge cases: empty set, universal set, disjoint sets
6. **Order Matters:** Remember $$A \times B \neq B \times A$$ and $$A - B \neq B - A$$

This comprehensive coverage includes all aspects of Set Theory with detailed explanations and examples, providing a solid foundation for solving MCQs and complex problems in the entrance examination.
