# π 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:**Β
Β Β - Sets β Capital letters: `A, B, C, ...`
Β Β - Elements β Small letters: `a, b, c, ...`
Β Β - `x β A` β x belongs to AΒ Β
Β Β - `x β A` β x does not belong to A
- Two Standard Forms :
Β Β | Form | Description | Example |
Β Β |---|---|---|
Β Β | **Roster/Tabular** | Elements listed inside `{}` separated by commas | `A = {2, 4, 6, 8}` |
Β Β | **Set-Builder** | Elements described by a common property: `{x : P(x)}` | `A = {x : x = 2n, n β β, n β€ 4}` |
> πΉ *Note:* In roster form, order & repetition do not matter: `{1, 2, 3} = {3, 1, 2, 1}`.
Β 2. Empty Set (Null / Void Set)
- **Definition:** A set containing **no elements**.
- **Notation:** `β ` or `{}`
- **Key Property:** `β β A` for **every** set A.
- **Examples:**Β
Β Β - `{x : xΒ² = β1, x β β}`
Β Β - `{x : x is a prime number between 8 and 10}`
- β οΈ `β β {0}` and `β β {β }`. `{0}` has 1 element; `{β }` has 1 element (the empty set itself).
Β 3. Finite and Infinite Sets
| Type | Definition | Cardinality `n(A)` | Example |
|---|---|---|---|
| **Finite** | Countable, definite number of elements | `n(A) = k` (k β ββ) | `A = {a, e, i, o, u} β n(A)=5` |
| **Infinite** | Uncountable or unlimited elements | Not a finite number | `β = {1,2,3,...}`, `β`, `β€` |
> πΉ `n(β ) = 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:**
Β Β - Order & repetition are ignored.
Β Β - `A = B β A β B` and `B β A`
- **Example:**Β
Β Β `A = {x : x is a letter in "MATH"}`Β Β
Β Β `B = {M, A, T, H}` β `A = B`
Β 5. Subsets
- **Definition:** `A` is a subset of `B` if every element of `A` is also in `B`.
- **Notation:** `A β B` (A is a subset of B)
- **Proper Subset:** `A β B` if `A β B` but `A β B` (at least one element in B is not in A)
- **Fundamental Properties:**
Β Β 1. `A β A` (Every set is a subset of itself)
Β Β 2. `β β A` (Empty set is subset of every set)
Β Β 3. If `A β B` and `B β C` β `A β C` (Transitivity)
Β Β 4. `A = B β A β B` and `B β A`
- **Number of Subsets:** If `n(A) = n`, then total subsets = `2βΏ`Β Β
Β Β *(Power set `P(A)` = set of all subsets of A)*
Β Β - Example: `A = {1, 2}` β Subsets: `β , {1}, {2}, {1,2}` β `2Β² = 4`
Β 6. Subsets of the Set of Real Numbers (Intervals)
The set of all real numbers is denoted by `β`. Many important subsets of `β` are expressed as **intervals**.
| Interval Type | Notation | Set-Builder Form | Endpoints Included? |
|---|---|---|---|
| **Open** | `(a, b)` | `{x β β : a < x < b}` | Neither `a` nor `b` |
| **Closed** | `[a, b]` | `{x β β : a β€ x β€ b}` | Both `a` and `b` |
| **Semi-open (Left)** | `(a, b]` | `{x β β : a < x β€ b}` | Only `b` |
| **Semi-open (Right)** | `[a, b)` | `{x β β : a β€ x < b}` | Only `a` |
πΉ Special Infinite Intervals
| Notation | Set-Builder Form | Meaning |
|---|---|---|
| `(a, β)` | `{x β β : x > a}` | All reals greater than `a` |
| `[a, β)` | `{x β β : x β₯ a}` | All reals β₯ `a` |
| `(-β, b)` | `{x β β : x < b}` | All reals less than `b` |
| `(-β, b]` | `{x β β : x β€ b}` | All reals β€ `b` |
| `(-β, β)` | `β` | Entire real number line |
πΉ Standard Number Sets (All are subsets of β)
- `β = {1, 2, 3, ...}` β Natural numbers
- `β€ = {..., -2, -1, 0, 1, 2, ...}` β Integers
- `β = {p/q : p, q β β€, q β 0}` β Rational numbers
- `T = β \ β` β Irrational numbers
- **Hierarchy:** `β β β€ β β β β` and `T β β`, with `β β© T = β `
π WBCHSE Exam Tips
1. Always write set notation precisely: `β, β, β, β, β , [, ], (, )`.
2. `β ` is a **subset** of every set, but an **element** only if explicitly placed inside braces.
3. When asked for subsets, remember to include `β ` and the set itself.
4. Interval questions often test bracket usage:Β Β
Β Β Β `[` or `]` β included (`β€` or `β₯`)Β Β
Β Β Β `(` or `)` β excluded (`<` or `>`)
5. In proofs, use: `A = B β A β B β§ B β A` and `A β B β A β B β§ A β B`.
β *Prepared as per WBCHSE Class 11 Mathematics Syllabus (Chapter: Sets)*Β Β
π *Recommended Practice:* NCERT/WBCHSE Textbook Ex. 1.1β1.3, solve previous years' 2-mark & 4-mark questions on set representation, subset counting, and interval conversion.