What is a set
Your school bag holds books, a box and a bottle. A cricket team is 11 players. In maths, such a clear group is called a set.
A set is a well-defined collection of objects. "Well-defined" means we can always say for sure whether an object belongs to it or not. Objects are its elements (members). We write 3 ∈ A for "3 belongs to A" and 5 ∉ A for "5 does not belong to A". Sets are named with capital letters.
Rule 1: Well-defined or not — Ask: can everybody decide the same answer for every object?
- Easy: The vowels of the English alphabet: a set (a, e, i, o, u).
- Medium: The tall students of your class: not a set — "tall" differs from person to person.
- Exam-type: The prime numbers below 20: a set; the delicious fruits in a market: not a set.
Rule 2: Use of ∈ and ∉ — Put the symbol between an object and the set.
- Easy: A = {2, 4, 6, 8}: 4 ∈ A and 5 ∉ A.
- Medium: B = {x : x is a letter of the word PAPA}: P ∈ B, but S ∉ B.
- Exam-type: Standard sets: N = natural numbers, Z = integers, Q = rational numbers, R = real numbers.
Order and repetition do not matter in a set: {1, 2, 3} and {3, 1, 2, 3} are the same set.
Treating "the best / tallest / cleverest" collections as sets. They are opinions, so they are not well-defined.
WELL = everyone agrees (no opinions).
Spot the difference
| Element | Set |
|---|---|
| A single member | The whole collection |
| Written with a small letter usually | Written with a capital letter |
| Symbol ∈ is used with it | Symbol { } is used around it |
Practice questions
- 1Is the collection of "good songs" a set?
- 2Is the collection of months starting with J a set?
- 3Write using ∈ or ∉: 9 ___ {1, 3, 5, 7}.
- 4How many elements does the set of letters of the word BALLOON have?
- 5Is {2, 3, 3, 5} the same set as {5, 3, 2}?
- 6Which one is a set: the days of a week, or the nicest days of a week?
- 7A = {x : x is a letter of the word MISSISSIPPI}. List its elements and say how many there are.
- 8Which of these belong to Z but not to N: −3, 0, 7, 12?
Show answers (try them yourself first)
- 1No — "good" is an opinion, so it is not well-defined.
- 2Yes — {January, June, July}.
- 39 ∉ {1, 3, 5, 7}
- 4The distinct letters are B, A, L, O, N, so 5.
- 5Yes — repetition and order are ignored.
- 6The days of a week.
- 7{M, I, S, P}, so 4 elements.
- 8−3 and 0 (N has only 1, 2, 3, …).
Roster form and set-builder form
To describe your friends you can either list names, or say "all students of your class who play kabaddi". Sets also have two ways.
In roster (tabular) form all elements are listed inside braces, separated by commas. In set-builder form a rule is given: A = {x : property of x}, read "the set of all x such that …".
Rule 1: Roster → set-builder — Find the pattern in the list and state it as a rule.
- Easy: {1, 3, 5, 7, 9} = {x : x is an odd natural number less than 10}.
- Medium: {3, 6, 9, 12} = {x : x = 3n, n ∈ N, n ≤ 4}.
- Exam-type: {1, 4, 9, 16, 25} = {x : x = n², n ∈ N, n ≤ 5}.
Rule 2: Set-builder → roster — Put each allowed value in the rule and list the results.
- Easy: {x : x is a natural number, x < 6} = {1, 2, 3, 4, 5}.
- Medium: {x : x ∈ N, x² < 30} = {1, 2, 3, 4, 5}.
- Exam-type: {x : x = 2n + 1, n ∈ {0, 1, 2, 3}} = {1, 3, 5, 7}.
Write each element once, even if the rule produces it several times.
Forgetting the condition on the variable (like x ∈ N) — the answer changes if x can be 0 or negative.
Roster = Register (list). Builder = Rule.
Spot the difference
| Roster form | Set-builder form |
|---|---|
| Lists every element | Gives a rule |
| Best for small sets | Best for large or infinite sets |
| {1, 2, 3} | {x : x ∈ N, x ≤ 3} |
Practice questions
- 1Write {x : x ∈ N, x ≤ 4} in roster form.
- 2Write {5, 10, 15, 20} in set-builder form.
- 3Write {x : x is a multiple of 4, x ≤ 20, x ∈ N} in roster form.
- 4Write {x : x = n² + 1, n ∈ {1, 2, 3}} in roster form.
- 5Write {x : x ∈ Z, −2 < x ≤ 3} in roster form.
- 6Write the set of letters in the word LEVEL in roster form.
- 7List the set {x : x ∈ N, x is a factor of 24}.
- 8List {x : x ∈ N, x² − 5x + 6 = 0}.
Show answers (try them yourself first)
- 1{1, 2, 3, 4}
- 2{x : x = 5n, n ∈ N, n ≤ 4}
- 3{4, 8, 12, 16, 20}
- 4{2, 5, 10}
- 5{-1, 0, 1, 2, 3}
- 6{L, E, V}
- 7{1, 2, 3, 4, 6, 8, 12, 24}
- 8x² − 5x + 6 = (x − 2)(x − 3), so {2, 3}.
Empty, finite and infinite sets
An empty lunch box has nothing in it, a school has a countable number of students, but the stars of the sky seem endless. Sets show the same three kinds.
The empty set (null set) has no element; it is written ∅ or { }. A set is finite if the elements can be counted and the counting ends, otherwise it is infinite. The number of elements of a finite set A is written n(A).
Rule 1: Spot the empty set — The condition can never be satisfied.
- Easy: {x : x ∈ N, 3 < x < 4} = ∅ (no natural number lies there).
- Medium: {x : x ∈ R, x² = −9} = ∅.
- Exam-type: {x : x is a prime number divisible by 6} = ∅.
Rule 2: Finite or infinite — Can you finish listing?
- Easy: The months of a year: finite, n = 12.
- Medium: The multiples of 5: infinite.
- Exam-type: The points on a line segment: infinite, even though the segment is short.
∅ is a set with zero elements, while {0} is a set with one element, the number 0.
Writing the empty set as {∅}. That set has one element (the empty set), so it is not empty.
∅ = empty plate. {0} = plate with one zero on it.
Spot the difference
| ∅ | {0} |
|---|---|
| No element | One element, 0 |
| n = 0 | n = 1 |
Practice questions
- 1What is n(∅)?
- 2Is {x : x ∈ N, x < 1} empty?
- 3Is the set of all days in a leap year finite?
- 4Is the set of integers between −3 and 3 (not included) finite? Find n.
- 5Is {x : x ∈ Z, x² = 2} empty?
- 6Is the set of all circles through a given point finite?
- 7Find n(A) for A = {x : x ∈ N, x is a factor of 36}.
- 8A has n(A) = 0 and B = {∅}. Find n(B).
Show answers (try them yourself first)
- 10
- 2Yes — the smallest natural number is 1.
- 3Yes, it has 366 elements.
- 4Yes: {-2, -1, 0, 1, 2}, so n = 5.
- 5Yes — no integer squares to 2.
- 6No — infinitely many.
- 7Factors: {1, 2, 3, 4, 6, 9, 12, 18, 36}, so n(A) = 9.
- 8n(B) = 1 because B has the element ∅.
Equal sets
Two baskets holding the same fruits are the same basket of fruits, whichever way you arrange them.
Two sets A and B are equal (A = B) if they have exactly the same elements: every element of A is in B and every element of B is in A.
Rule 1: Test equality — List both sets in roster form (each element once) and compare.
- Easy: {1, 2, 3} and {3, 2, 1}: equal.
- Medium: A = {x : x ∈ N, x² < 10} = {1, 2, 3} and B = {1, 2, 3}: equal.
- Exam-type: {x : x² − 4 = 0, x ∈ R} = {2, −2} but {x : x ∈ N, x² = 4} = {2}: not equal.
Rule 2: Find an unknown — Match elements and solve.
- Easy: {2, x} = {2, 7}: x = 7.
- Medium: {3, x + 1} = {3, 9}: x = 8.
- Exam-type: {x, 5} = {y, 5} and x ≠ 5: then y = x.
Equal sets must have the same number of elements; sets with equal numbers of elements need not be equal.
Thinking {1, 2} and {3, 4} are equal because both have 2 elements.
EQUAL = same elements, not just same count.
Practice questions
- 1Are {a, b, c} and {c, a, b} equal?
- 2Are {1, 2} and {2, 3} equal?
- 3If {4, y} = {4, 10}, find y.
- 4Is {x : x ∈ N, x < 5} = {1, 2, 3, 4}?
- 5Are {x : x is a letter of PEEP} and {P, E} equal?
- 6Are {x : x ∈ Z, x² = 9} and {3} equal?
- 7If {1, x + 2, 9} = {1, 5, 9}, find x.
- 8Is {x : x ∈ N, x is even and prime} = {2}?
Show answers (try them yourself first)
- 1Yes.
- 2No — 1 is only in the first.
- 3y = 10
- 4Yes.
- 5Yes.
- 6No — −3 is also in the first set.
- 7x + 2 = 5, so x = 3.
- 8Yes — 2 is the only even prime.