Sets & Intervals – Complete Notes for Class 11, CUET, JKSSB & NDA
What You'll Learn
- All 4 types of intervals with notation and examples
- How to write intervals in set builder form
- Power Set — definition, formula, and worked examples
- Universal Set — meaning and context-based usage
- Complete comparison table of all set types
- PYQ-style practice questions with answers
Why Sets & Intervals Are the Most Important Foundation in Class 11 Maths
Sets and Intervals are the first chapter in Class 11 Mathematics, and for good reason — they are the language in which the rest of mathematics is written. Every topic that follows — Relations, Functions, Probability, Sequences, Calculus — uses set notation and interval representation. Students who understand this chapter deeply find the rest of Class 11 significantly easier.
From an exam perspective, this topic appears in CUET Mathematics, JKSSB quantitative aptitude sections, NDA Mathematics Paper, and JEE Mains. Questions are typically direct and formula-based — making this one of the most reliable chapters to score full marks from.
1. Intervals as Subsets of Real Numbers
An interval is a continuous subset of real numbers lying between two given boundary values. If a and b are real numbers with a < b, we can define four types of intervals depending on whether the endpoints are included or excluded.
The key symbols to remember: round brackets ( ) mean the endpoint is excluded, and square brackets [ ] mean the endpoint is included.
Both endpoints excluded. Round brackets on both sides.
Both endpoints included. Square brackets on both sides.
a excluded, b included. Open on left, closed on right.
a included, b excluded. Closed on left, open on right.
Infinite Intervals
When an interval extends indefinitely in one or both directions, we use the infinity symbol (∞). The critical rule is that infinity is never included — it is always written with a round bracket.
| Notation | Set Builder Form | Meaning |
|---|---|---|
| (a, ∞) | { x ∈ R : x > a } | All reals greater than a |
| [a, ∞) | { x ∈ R : x ≥ a } | All reals greater than or equal to a |
| (-∞, b) | { x ∈ R : x < b } | All reals less than b |
| (-∞, b] | { x ∈ R : x ≤ b } | All reals less than or equal to b |
| (-∞, ∞) | { x ∈ R } | The entire real number line |
2. Writing Intervals in Set Builder Form
Converting between interval notation and set builder form is a very commonly tested skill. The key is to identify the bracket type and the number system (Real numbers R, Integers Z, or Natural numbers N).
Worked Examples
Example 1: Write (-6, 12] in set builder form over real numbers.
Answer: { x ∈ R : -6 < x ≤ 12 }
Example 2: Write (-4, 4) in set builder form over integers only.
Answer: { x ∈ Z : -4 < x < 4 } = {-3, -2, -1, 0, 1, 2, 3}
Example 3: Write [0, ∞) in set builder form.
Answer: { x ∈ R : x ≥ 0 } — this is the set of all non-negative real numbers.
3. Power Set
The Power Set of a set A, written as P(A), is the set that contains all possible subsets of A — including the empty set (∅) and A itself.
Example: Finding Power Set of A = {1, 2, 3}
First list all subsets of A systematically:
0-element subsets: ∅
1-element subsets: {1}, {2}, {3}
2-element subsets: {1,2}, {1,3}, {2,3}
3-element subsets: {1,2,3}
Therefore: P(A) = { ∅, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3} }
Total subsets = 8 = 2³ ✓
The Key Formulas
Quick Reference Table
| n (elements) | Total Subsets (2^n) | Proper Subsets (2^n−2) | Non-empty Subsets (2^n−1) |
|---|---|---|---|
| 0 | 1 | −1 (not applicable) | 0 |
| 1 | 2 | 0 | 1 |
| 2 | 4 | 2 | 3 |
| 3 | 8 | 6 | 7 |
| 4 | 16 | 14 | 15 |
| 5 | 32 | 30 | 31 |
4. Universal Set
The Universal Set (U) is the master set that contains all elements being considered in a particular problem or context. Every other set in the discussion is a subset of U. It is also called the superset in that context.
The universal set is not fixed — it changes depending on what the problem is about. If we are discussing even numbers, U might be all natural numbers. If we are discussing letters in the word "MATHS," U = {M, A, T, H, S}.
Example
If A = {1, 2}, B = {2, 3, 4}, and C = {3, 4, 5}, then a suitable universal set is:
U = {1, 2, 3, 4, 5}
The complement of A (written A' or Aᶜ) is everything in U that is NOT in A:
A' = U − A = {3, 4, 5}
5. Complete Comparison Table
| Concept | Symbol | Definition | Key Property |
|---|---|---|---|
| Open Interval | (a, b) | a < x < b | Endpoints excluded |
| Closed Interval | [a, b] | a ≤ x ≤ b | Endpoints included |
| Half-Open | (a,b] or [a,b) | One endpoint each | Mixed inclusion |
| Infinite Interval | (a, ∞) | x > a | ∞ never included |
| Power Set | P(A) | Set of all subsets | n(P(A)) = 2^n |
| Universal Set | U | Contains all elements in context | Context-dependent |
| Complement | A' | U − A | A ∪ A' = U |
📋 Practice Questions (CUET / JKSSB / NDA Pattern)
These follow the exact question pattern of CUET Mathematics, JKSSB quantitative aptitude, and NDA Maths papers. Try solving before checking answers.
Write [2, 7) in set builder form.
✓ Answer: { x ∈ R : 2 ≤ x < 7 }How many subsets does a set with 5 elements have?
✓ Answer: (c) 32How many proper subsets does a set with 4 elements have?
✓ Answer: (b) 14Write the power set of A = {a, b}.
✓ Answer: P(A) = { ∅, {a}, {b}, {a,b} }Express (-∞, 5] in set builder form.
✓ Answer: { x ∈ R : x ≤ 5 }If U = {1,2,3,4,5} and A = {1,3,5}, find A' (complement of A).
✓ Answer: (b) {2, 4}Can infinity ever be included in an interval using square brackets?
✓ Answer: (c)If n(A) = 3, find n(P(A)).
✓ Answer: (c) 8Common Mistakes Students Make
Including infinity in square brackets: Writing [a, ∞] is mathematically incorrect. Infinity is not a number and cannot be an endpoint of a closed interval. Always write [a, ∞) or (-∞, b].
Forgetting the empty set in the power set: The empty set ∅ is a subset of every set. It must always be included in P(A). Students who list subsets systematically rarely make this mistake.
Confusing subset and proper subset: Every set is a subset of itself, but NOT a proper subset of itself. The empty set is both a subset and a proper subset of any non-empty set.
Wrong formula for proper subsets: Many students use 2^n − 1 (which gives non-empty subsets) when asked for proper subsets. The correct formula is 2^n − 2 (excluding empty set AND the full set).
Forgetting domain when writing set builder form: (-4, 4) over integers ≠ (-4, 4) over reals. Always state whether x ∈ R, x ∈ Z, or x ∈ N.
⚡ Quick Revision Cheatsheet
Final Thoughts
Sets and Intervals are not just a standalone chapter — they are the vocabulary of mathematics. Every concept you'll encounter in Class 11 and 12 Maths, from Functions and Relations to Probability and Calculus, uses this notation. Investing time in mastering this chapter now will pay dividends throughout your entire preparation.
The good news is that questions from this chapter in CUET, JKSSB, and NDA are almost always direct and formulaic. If you remember the four interval types, the 2^n formula, and the difference between proper and improper subsets — you will answer every exam question correctly.