Set 

# 📘 Class 11 Mathematics (WBCHSE) | Chapter: Sets | Revision Notes

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1. Sets and Their Representations

- **Definition:** A set is a *well-defined* collection of distinct objects. "Well-defined" means there is no ambiguity in deciding whether an object belongs to the set.

- **Notation:** 

  - Sets → Capital letters: `A, B, C, ...`

  - Elements → Small letters: `a, b, c, ...`

  - `x ∈ A` → x belongs to A  

  - `x ∉ A` → x does not belong to A

- Two Standard Forms :

  | Form | Description | Example |

  |---|---|---|

  | **Roster/Tabular** | Elements listed inside `{}` separated by commas | `A = {2, 4, 6, 8}` |

  | **Set-Builder** | Elements described by a common property: `{x : P(x)}` | `A = {x : x = 2n, n ∈ ℕ, n ≤ 4}` |

> 🔹 *Note:* In roster form, order & repetition do not matter: `{1, 2, 3} = {3, 1, 2, 1}`.

 2. Empty Set (Null / Void Set)

- **Definition:** A set containing **no elements**.

- **Notation:** `∅` or `{}`

- **Key Property:** `∅ ⊆ A` for **every** set A.

- **Examples:** 

  - `{x : x² = –1, x ∈ ℝ}`

  - `{x : x is a prime number between 8 and 10}`

- ⚠️ `∅ ≠ {0}` and `∅ ≠ {∅}`. `{0}` has 1 element; `{∅}` has 1 element (the empty set itself).

 3. Finite and Infinite Sets

| Type | Definition | Cardinality `n(A)` | Example |

|---|---|---|---|

| **Finite** | Countable, definite number of elements | `n(A) = k` (k ∈ ℕ₀) | `A = {a, e, i, o, u} → n(A)=5` |

| **Infinite** | Uncountable or unlimited elements | Not a finite number | `ℕ = {1,2,3,...}`, `ℝ`, `ℤ` |

> 🔹 `n(∅) = 0` (Finite set with zero elements)

4. Equal Sets

- **Definition:** Two sets `A` and `B` are equal if they contain **exactly the same elements**.

- **Notation:** `A = B`

- **Properties:**

  - Order & repetition are ignored.

  - `A = B ⇔ A ⊆ B` and `B ⊆ A`

- **Example:** 

  `A = {x : x is a letter in "MATH"}`  

  `B = {M, A, T, H}` → `A = B`

 5. Subsets

- **Definition:** `A` is a subset of `B` if every element of `A` is also in `B`.

- **Notation:** `A ⊆ B` (A is a subset of B)

- **Proper Subset:** `A ⊂ B` if `A ⊆ B` but `A ≠ B` (at least one element in B is not in A)

- **Fundamental Properties:**

  1. `A ⊆ A` (Every set is a subset of itself)

  2. `∅ ⊆ A` (Empty set is subset of every set)

  3. If `A ⊆ B` and `B ⊆ C` ⇒ `A ⊆ C` (Transitivity)

  4. `A = B ⇔ A ⊆ B` and `B ⊆ A`

- **Number of Subsets:** If `n(A) = n`, then total subsets = `2ⁿ`  

  *(Power set `P(A)` = set of all subsets of A)*

  - Example: `A = {1, 2}` → Subsets: `∅, {1}, {2}, {1,2}` → `2² = 4`

 6. Subsets of the Set of Real Numbers (Intervals)

The set of all real numbers is denoted by `ℝ`. Many important subsets of `ℝ` are expressed as **intervals**.

| Interval Type | Notation | Set-Builder Form | Endpoints Included? |

|---|---|---|---|

| **Open** | `(a, b)` | `{x ∈ ℝ : a < x < b}` | Neither `a` nor `b` |

| **Closed** | `[a, b]` | `{x ∈ ℝ : a ≤ x ≤ b}` | Both `a` and `b` |

| **Semi-open (Left)** | `(a, b]` | `{x ∈ ℝ : a < x ≤ b}` | Only `b` |

| **Semi-open (Right)** | `[a, b)` | `{x ∈ ℝ : a ≤ x < b}` | Only `a` |

🔹 Special Infinite Intervals

| Notation | Set-Builder Form | Meaning |

|---|---|---|

| `(a, ∞)` | `{x ∈ ℝ : x > a}` | All reals greater than `a` |

| `[a, ∞)` | `{x ∈ ℝ : x ≥ a}` | All reals ≥ `a` |

| `(-∞, b)` | `{x ∈ ℝ : x < b}` | All reals less than `b` |

| `(-∞, b]` | `{x ∈ ℝ : x ≤ b}` | All reals ≤ `b` |

| `(-∞, ∞)` | `ℝ` | Entire real number line |

🔹 Standard Number Sets (All are subsets of ℝ)

- `ℕ = {1, 2, 3, ...}` → Natural numbers

- `ℤ = {..., -2, -1, 0, 1, 2, ...}` → Integers

- `ℚ = {p/q : p, q ∈ ℤ, q ≠ 0}` → Rational numbers

- `T = ℝ \ ℚ` → Irrational numbers

- **Hierarchy:** `ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ` and `T ⊂ ℝ`, with `ℚ ∩ T = ∅`

📝 WBCHSE Exam Tips

1. Always write set notation precisely: `∈, ∉, ⊆, ⊂, ∅, [, ], (, )`.

2. `∅` is a **subset** of every set, but an **element** only if explicitly placed inside braces.

3. When asked for subsets, remember to include `∅` and the set itself.

4. Interval questions often test bracket usage:  

   `[` or `]` → included (`≤` or `≥`)  

   `(` or `)` → excluded (`<` or `>`)

5. In proofs, use: `A = B ⇔ A ⊆ B ∧ B ⊆ A` and `A ⊂ B ⇔ A ⊆ B ∧ A ≠ B`.

✅ *Prepared as per WBCHSE Class 11 Mathematics Syllabus (Chapter: Sets)*  

📖 *Recommended Practice:* NCERT/WBCHSE Textbook Ex. 1.1–1.3, solve previous years' 2-mark & 4-mark questions on set representation, subset counting, and interval conversion.