A set is a collection of things, and a Venn diagram shows how sets overlap. Learn the notation, fill diagrams from given counts, and (Extended) work with three sets and subsets.
Fill a Venn diagram from the middle outwards: the intersection first, then the 'only' regions, then the outside. \(n(A \cup B) = n(A) + n(B) - n(A \cap B)\), because the overlap is counted twice otherwise.
Fig. 1: ξ = {1, 2, ..., 12}, A = {even numbers}, B = {multiples of 3}. The overlap A ∩ B = {6, 12}; the numbers outside both circles form (A ∪ B)′ = {1, 5, 7, 11}
Section 01
Set language.
In the lab, set the universal set \(\xi = \{1, \dots, n\}\), choose a rule for A and for B, and watch each number drop into its region of the Venn diagram. Then pick a region and read off its name in set notation.
Notation
Meaning
\(\xi\)
the universal set: everything being considered
\(n(A)\)
the number of elements in A
\(A \cup B\)
union: in A or B or both
\(A \cap B\)
intersection: in both A and B
\(A'\)
complement: in \(\xi\) but not in A
\(x \in A\), \(x \notin A\)
(Extended) \(x\) is / is not an element of A
\(\varnothing\)
(Extended) the empty set
\(A \subseteq B\), \(A \not\subseteq B\)
(Extended) A is / is not a subset of B: every element of A is also in B
Worked example Listing sets
\(\xi = \{1, 2, 3, \dots, 12\}\), \(A = \{\text{even numbers}\}\), \(B = \{\text{multiples of 3}\}\). List \(A \cap B\), \(A \cup B\) and \(A'\), and find \(n(A \cup B)\).
Intersection: the numbers in both lists, \(A \cap B = \{6, 12\}\).
Union: every number in either list, written once: \(A \cup B = \{2, 3, 4, 6, 8, 9, 10, 12\}\), so \(n(A \cup B) = 8\).
Complement: the numbers in \(\xi\) not in A, \(A' = \{1, 3, 5, 7, 9, 11\}\).
Answer: \(A \cap B = \{6, 12\}\), \(A \cup B = \{2, 3, 4, 6, 8, 9, 10, 12\}\), \(A' = \{1, 3, 5, 7, 9, 11\}\), \(n(A \cup B) = 8\) (see Fig. 1)
Worked example Set-builder notation
List the elements of \(C = \{x : x \text{ is an integer},\ 2 \le x < 7\}\).
Read it as: the set of all \(x\) such that \(x\) is an integer from 2 up to, but not including, 7.
\(\le\) includes 2; \(<\) excludes 7.
Answer: \(C = \{2, 3, 4, 5, 6\}\)
Examiner trap
In a union, list a shared element only ONCE. Writing \(A \cup B = \{2, 4, 6, 8, 10, 12, 3, 6, 9, 12\}\) double-counts 6 and 12 and gives the wrong \(n(A \cup B)\).
Section 02
Venn diagrams from counts.
Often you are given numbers of elements, not the elements themselves. Put the intersection in first, subtract to find the 'only' regions, and use the total to find what is outside.
Worked example Finding the overlap
In a group of 30 students, 18 study art (A), 15 study music (M) and 4 study neither. How many study both? How many study art only?
Students in at least one subject: \(30 - 4 = 26\), so \(n(A \cup M) = 26\).
\(n(A) + n(M) = 18 + 15 = 33\), which counts the 'both' students twice.
Do not write \(n(A) = 18\) in the 'A only' region. 18 is the whole circle; the 'A only' region is \(18 - 7 = 11\). Writing 18 there makes the diagram add up to more than 30.
Stage 1 · Learn
Check your understanding
Four quick checks. Each correct answer earns XP and records this skill on your profile.
\(A = \{2, 4, 6, 8\}\) and \(B = \{4, 8, 12\}\). Which is \(A \cap B\)?
Quick check+10 XP
Which notation means 'in A or in B or in both'?
Quick check+10 XP
(Extended) \(A = \{1, 3\}\) and \(B = \{1, 2, 3, 4\}\). Which statement is true?
Section 03
Three sets and subsets.
Extended. Venn diagrams can have three overlapping circles, with eight regions in all. Work from the centre (in all three) outwards, one region at a time.
\(\{2, 4\} \subseteq A\) because 2 and 4 are both even; \(\{2, 3\} \not\subseteq A\) because 3 is odd.
\(A \cap B' \) means 'in A but not in B': for Fig. 1 it is \(\{2, 4, 8, 10\}\).
If two sets do not overlap, their intersection is the empty set: \(\{\text{odd numbers}\} \cap \{\text{even numbers}\} = \varnothing\).
Worked example Three sets (Extended)
\(\xi = \{1, 2, \dots, 12\}\), \(A = \{\text{even numbers}\}\), \(B = \{\text{multiples of 3}\}\), \(C = \{\text{factors of 12}\}\). Find \(A \cap B \cap C\), list \(A \cap C \cap B'\), and find \(n\big((A \cup B \cup C)'\big)\).