Sets and Their Representations

Sets are the alphabet of higher mathematics — relations, functions, probability and calculus are all built on them. This chapter opens with the single idea everything rests on: a set is a well-defined collection of distinct objects, called its elements.

What "well-defined" means

A collection is a set only if, for any object, you can decide unambiguously whether it belongs. "Vowels in the English alphabet" is a set; "tall students of a class" is not (tall is subjective).

Membership: if a is an element of set A we write a ∈ A; if not, a ∉ A.

Two ways to describe a set

• Roster (tabular) form: list every element inside braces, separated by commas — e.g. A = {2, 3, 5, 7}. Order does not matter and no element is repeated.
• Set-builder form: state the common property — e.g. A = {x : x is a prime number less than 10}. Read the colon as "such that".

Standard number sets (memorise the symbols)

N = natural numbers {1, 2, 3, …}; W = whole numbers {0, 1, 2, …}; Z = integers; Q = rationals (p/q, q ≠ 0); R = real numbers; T = irrationals (R − Q); C = complex numbers. Note N ⊂ W ⊂ Z ⊂ Q ⊂ R.

Exam Tricks & Tips

• 🎯 In roster form never repeat an element: the letters of "SCHOOL" give {S, C, H, O, L}, only 5 elements.
• 🎯 Order is irrelevant: {1, 2, 3} = {3, 1, 2} — the same set.
• 🎯 Convert fluently both ways; questions often give one form and ask for the other.
• 🎯 ∈ links an element to a set; ⊂ links a set to a set — never mix them.
• 🎯 Memorise N ⊂ W ⊂ Z ⊂ Q ⊂ R and that √2, π ∈ T (irrationals), not Q.
• ❌ Common mistake: writing "collection of good players" as a set — it is not well-defined, so it fails the very definition.

Expected exam pattern

Direct 1-mark questions: convert between forms, decide membership, or spot the non-well-defined collection. Foundational — every later topic assumes this fluency.

Cover the answer, recall, then check. 11 cards on the basics of sets.

Q1. What is a set?
A1. A well-defined collection of distinct objects (its elements).

Q2. What does "well-defined" require?
A2. For any object you can decide unambiguously whether it belongs. "Vowels" qualifies; "tall students" does not.

Q3. Write the set of letters of the word "SCHOOL" in roster form.
A3. {S, C, H, O, L} — 5 elements; repeats (O) are written once.

Q4. Does order matter in a set? Is {1,2,3} = {3,2,1}?
A4. Order does not matter; yes, they are the same set.

Q5. Difference between ∈ and ⊂?
A5. ∈ relates an element to a set (2 ∈ N); ⊂ relates a set to a set (N ⊂ Z).

Q6. Give A = {x : x is a prime < 10} in roster form.
A6. {2, 3, 5, 7}.

Q7. What do N, W, Z, Q, R stand for?
A7. Natural, Whole, Integers, Rationals, Reals.

Q8. Which set contains 0 but not negatives: N or W?
A8. W (whole numbers); N starts at 1.

Q9. Is √2 rational? Which standard set is it in?
A9. No — it is irrational (T = R − Q).

Q10. State the chain of inclusions among number sets.
A10. N ⊂ W ⊂ Z ⊂ Q ⊂ R.

Q11. Convert {x : x = n², n ∈ N, n ≤ 4} to roster form.
A11. {1, 4, 9, 16}.

The idea that organises all of mathematics

Every branch of higher maths — functions, probability, geometry — is built on one deceptively simple object: the set. Master this, and the rest of Class 11 clicks into place.

Core definition: A set is a well-defined collection of distinct objects. "Well-defined" means there is a clear, unambiguous rule to decide whether any given object belongs. "The tall students of a class" is NOT a set (tall is vague); "the students taller than 165 cm" IS a set.

The objects in a set are its elements (or members). We write a ∈ A ("a belongs to A") or a ∉ A ("a does not belong to A").

Building understanding, layer by layer

Beginner: what counts as a set

Sets are written with capital letters (A, B, N, Z) and elements inside braces { }. Two golden rules:

• Distinctness: repetition is ignored. {1, 2, 2, 3} is just {1, 2, 3}.
• Order is irrelevant: {a, b, c} = {c, a, b}.

Intermediate: the two ways to represent a set

1. Roster (tabular) form — list every element between braces: A = {2, 4, 6, 8}.
2. Set-builder form — state the defining property: A = {x : x is an even natural number, x ≤ 8}. Read the colon ":" as "such that".

Set-builder shines when the set is large or infinite: {x : x ∈ ℕ, x is prime} can never be fully listed, but the rule captures it perfectly.

Advanced: the standard number sets

Memorise these symbols — they appear on every exam:

• ℕ = natural numbers {1, 2, 3, …}
• W = whole numbers {0, 1, 2, …}
• ℤ = integers {…, −2, −1, 0, 1, 2, …}
• ℚ = rationals {p/q : p, q ∈ ℤ, q ≠ 0}
• ℝ = real numbers

Nesting: ℕ ⊂ W ⊂ ℤ ⊂ ℚ ⊂ ℝ.

Worked example

Q. Write A = {x : x is an integer and x² < 20} in roster form.
Solution. We need integers whose square is below 20. Test outward from 0: 0²=0, (±1)²=1, (±2)²=4, (±3)²=9, (±4)²=16 (all < 20), (±5)²=25 (too big). So
A = {−4, −3, −2, −1, 0, 1, 2, 3, 4}.
Note we include negatives — a very common slip is to list only positive integers.

Where it shows up

Databases, search filters, and spreadsheet queries are literally set-builder conditions. When you write "SELECT name WHERE age > 18", you are defining a set.

Exam tricks & shortcuts

• To convert set-builder → roster, test small values outward from 0 until the condition fails on both sides.
• Mnemonic for the number chain "N-W-Z-Q-R": "Nine Wise Zebras Question Reality" — the order of increasing size.
• ℤ comes from the German Zahlen (numbers); ℚ from Quotient — knowing this cements which symbol is which.

Key formulas / notation

• Roster form: list elements, e.g. A = {1, 2, 3}.
• Set-builder form: A = {x : P(x)} (x such that property P holds).
• Membership: x ∈ A, x ∉ A.
• Standard number sets: ℕ (naturals), ℤ (integers), ℚ (rationals), ℝ (reals), ℂ (complex).
• Cardinality: n(A) = number of distinct elements.
• Empty set: ∅ (n = 0); singleton: one element.

A set is a well-defined collection of distinct objects; order and repetition do not matter.

Worked Example

Problem: Write the set A = { x : x is a natural number and x² < 20 } in roster form.

Solution:
Step 1 — Understand the condition. We need natural numbers x (i.e. 1, 2, 3, …) whose square is less than 20.

Step 2 — Test natural numbers one by one:
1² = 1 < 20 ✓; 2² = 4 < 20 ✓; 3² = 9 < 20 ✓; 4² = 16 < 20 ✓.

Step 3 — Check the next value:
5² = 25, which is not less than 20, so 5 and all larger numbers are excluded.

Step 4 — List the qualifying elements in roster (tabular) form:
A = {1, 2, 3, 4}.

Answer: A = {1, 2, 3, 4}.