Sets
Set notation: listing, union, intersection, complement
In this note
- 1.2Use set notation: \(n(A)\), \(\in\), \(\notin\), \(A'\), \(\varnothing\), \(\xi\), \(\subseteq\), \(A\cup B\), \(A\cap B\).
- 1.2List a set described in words or in set-builder notation, e.g. \(\{x: 2\le x<6\}\).
- 1.2Find unions, intersections and complements, and count their elements.
Next: shading Venn diagrams and Venn diagram problems.
Formula bank
Sets and their symbols
A set is a collection of elements, each listed once, in any order. The universal set \(\xi\) is everything the question is about; every other set lives inside it.
| Symbol | Meaning | Example (ξ = {1, …, 15}, A = {3, 6, 9, 12, 15}) |
|---|---|---|
| \(n(A)\) | number of elements | \(n(A)=5\) |
| \(\in\) / \(\notin\) | is / is not an element of | \(9\in A\), \(4\notin A\) |
| \(A'\) | not in A | \(A'=\{1,2,4,5,7,8,10,11,13,14\}\) |
| \(\varnothing\) | the empty set | { } |
| \(\{x: 2\le x<6\}\) | all x with 2 ≤ x < 6 | \(\{2,3,4,5\}\) (integers) |
Try it yourself
Show answer
Answer \(\{2,3,4,5\}\)
Show answer
Answer \(\{0,1,2,3\}\)
Show answer
Answer 10
Union, intersection and complement
General methodHow to answer any set-listing question
- List every set from ξ first (leave out anything not in ξ).
- ∩ (AND): elements in BOTH lists. ∪ (OR): elements in EITHER, each once. ′ (NOT): ξ minus the set.
- Combinations like \(A\cap B'\): in A AND not in B.
- n( ) asks for a number: count the list.
Example ξ = {1, …, 15}, A = {multiples of 3}, B = {factors of 30}. Find \(A\cap B\) and \(n(A\cup B)\).
List both from ξ, then compare.
- \(A=\{3,6,9,12,15\}\), \(B=\{1,2,3,5,6,10,15\}\) (30 is not in ξ)
- \(A\cap B=\{3,6,15\}\)
- \(A\cup B=\{1,2,3,5,6,9,10,12,15\}\)
Answer \(\{3,6,15\}\); 9
Try it yourself
Show answer
Answer \(\{4,7,8,11,13,14\}\)
Show answer
Answer \(\{9,12\}\)
Ways they test it
Type 1 List the elements
"List the elements of \(P\cap Q\)." (1 mark)
Spot it "list".
- List each set from ξ.
- Apply ∩ / ∪ / ′.
Type 2 Count
"Find \(n(A\cup B)\)." (1 mark)
Spot it "n(…)".
- List, then count (shared ones once).
Type 3 A possible element
"\(y\in E\cap M\). Write down a possible value of y."
Spot it "a possible value".
- Find one element in the set described.
For example ξ = {1, …, 20}, E even, M multiples of 5
- E ∩ M = {10, 20}
Answer 10
Type 4 Subsets and symbols
"Write down a subset of A with 2 elements." / "Is \(\{3,9\}\subseteq A\)?"
Spot it ⊆, ∈, ∉.
- Check every element.
Mistakes to avoid
- ✗ Including elements outside ξ (30 in a set from 1 to 15)✓ only ξ's elementsξ is everything.
- ✗ n(A ∪ B) = n(A) + n(B)✓ subtract the overlapShared elements count once.
- ✗ A list given for n(…)✓ a numbern means how many.
- ✗ Mixing up ∪ and ∩✓ ∪ = OR, ∩ = AND∪ is a cup holding everything.
Summary
- List every set from ξ first.
- ∪ = OR (either), ∩ = AND (both), ′ = NOT.
- n( ) is a count; list each element once.
- \(n(A\cup B)=n(A)+n(B)-n(A\cap B)\).
If you see …
- "list"curly brackets with the elements.
- "n(…)"a number.
Practice
Practice: easy, medium, hard
Easy
Show working
- A′ = {2, 4, 6, 8, 10}
Answer 5
Show working
- each element once
Answer \(\{1,2,3,4\}\)
Show working
- in both
Answer \(\{2,3\}\)
Medium
Guide me
What is it asking? P ∩ Q
What do we know? P = {2, 3, 5, 7, 11}, Q = {2, 3, 4, 6, 8, 12}
Plan in both
Show working
- 2 and 3
Answer \(\{2,3\}\)
Guide me
What is it asking? a count
What do we know? 5 + 6 − 2
Plan shared once
Show working
- \(5+6-2\)
Answer 9
Guide me
What is it asking? the complement
What do we know? ξ minus the union
Plan what is left
Show working
- 9 and 10 are in neither
Answer \(\{9,10\}\)
Hard
Guide me
What is it asking? a count
What do we know? F = {1,2,3,4,6,12}, O = {1,3,5,7,9,11}
Plan 6 + 6 − 2
Show working
- F ∩ O = {1, 3}
- \(6+6-2\)
Answer 10
| Part | Answer / working | Marks | Guidance |
|---|---|---|---|
| F and O listed | M1 | ||
| 10 | A1 | ||
| Total: 2 marks | |||
Guide me
What is it asking? E ∩ M
What do we know? even AND a multiple of 5
Plan multiples of 10
Show working
- 10, 20
Answer \(\{10,20\}\)
| Part | Answer / working | Marks | Guidance |
|---|---|---|---|
| \(\{10,20\}\) | B1 | ||
| Total: 1 mark | |||
Guide me
What is it asking? in A, not in B
What do we know? A = {3,6,9,12,15}
Plan remove those also in B
Show working
- 3, 6, 15 are in B
Answer \(\{9,12\}\)
| Part | Answer / working | Marks | Guidance |
|---|---|---|---|
| \(\{9,12\}\) | B1 | ||
| Total: 1 mark | |||
Can I do this?
- I know every set symbol and can read set-builder notation.
- I can list unions, intersections and complements.
- I can find n( ) of any combination of sets.
Anything unticked: revise that section first. All ticked: take the quiz.
Quiz
The multiple-choice and guided questions need JavaScript. The concept checks and exam practice below work without it.
Your progress on this note is saved when JavaScript is on.