SETS — ONE-PAGE REVISION NOTES

Class XI | CBSE

🔹 1. What is a Set?

A set is a well-defined collection of objects.

✔ Well-defined → No ambiguity

❌ Not well-defined → Depends on opinion

Example (Set):

Natural numbers less than 10

Not a Set:

Best students of a class

2. Important Standard Sets

\[ \begin{aligned} N &:\; \text{Natural numbers} \\ Z &:\; \text{Integers} \\ Q &:\; \text{Rational numbers} \\ R &:\; \text{Real numbers} \\ Z^{+} &:\; \text{Positive integers} \\ Q^{+} &:\; \text{Positive rational numbers} \\ R^{+} &:\; \text{Positive real numbers} \end{aligned} \]

🔹 3. Elements & Symbols

• Elements / members → same meaning
• Sets → Capital letters (A, B, C)
• Elements → Small letters (a, b, x)

Symbols

\[ \begin{aligned} \in &:\; \text{belongs to} \\ \notin &:\; \text{does not belong to} \end{aligned} \]

Example: \[ a \in \{a, e, i, o, u\}, \quad b \notin \{a, e, i, o, u\} \]

🔹 4. Representation of Sets

(i) Roster (Tabular) Form

• Elements listed inside { }
• Order does not matter
• No repetition

Example: \[ \{2, 4, 6\} \]

(ii) Set-Builder Form

General form:

\[ \{x : \text{property of } x\} \]

Example:

\[ \{x : x \in N,\; x < 6\} \]

5. Conversion Examples

Roster → Set-Builder

\[ \{1, 4, 9, 16\} \]

\[ \{x : x = n^{2},\; n \in N,\; 1 \le n \le 4\} \]

Set-Builder → Roster

\[ \{x : x \in Z,\; -2 \le x \le 3\} \]

\[ \{-2, -1, 0, 1, 2, 3\} \]

🔹 6. Key Exam Points (VERY IMPORTANT)

• ✔ Order of elements does not matter
• ✔ Repetition is not allowed
• ✔ Always mention domain (N, Z, etc.)
• ✔ Words like best, famous, dangerous → ❌ Not a set
• ✔ Write Math step-by-step

7. Common Mistakes to Avoid

• ❌ Writing repeated elements
• ❌ Forgetting domain (N / Z)
• ❌ Treating subjective collections as sets
• ❌ Writing long equations in one line

EXAM QUICK CHECK

1. Is the collection well-defined?
2. Are elements distinct?
3. Is the representation correct?