Using Set Notation
Every question with a fully worked solution.
- Using Set Notation - Video - Venn Diagrams with set notation Watch
Theory
Set notation is a short way to name the regions of a Venn diagram: \(A \cap B\) (and), \(A \cup B\) (or), \(A'\) (not) and \(n(A)\) (how many). This Year 9 maths page shows how to read and write the notation, list sets of numbers, find a missing region, move between a Venn diagram and a two-way table, and calculate probabilities, with counts or with probabilities in the regions.
A set is a collection of distinct objects. The universal set \(\xi\) (or \(S\)) holds everything and is drawn as a rectangle. Each event is a circle inside it.
\(A \cap B\), the intersection, means "\(A\) and \(B\)": the overlap. \(A \cup B\), the union, means "\(A\) or \(B\) or both": everything inside the circles.
\(A'\) (also written \(\overline{A}\) or \(A^c\)), the complement of \(A\), means "not \(A\)": everything outside circle \(A\).
\(n(A)\) means the number of elements in \(A\).
Events are mutually exclusive if they cannot happen together. Their circles do not overlap, and \(P(A \cup B) = P(A) + P(B)\).
The addition rule (the overlap is counted twice when you add the circles, so subtract it once):
| In words | Notation | Regions |
|---|---|---|
| \(A\) and \(B\) | \(A\cap B\) | the overlap |
| \(A\) or \(B\) (or both) | \(A\cup B\) | all three regions inside the circles |
| \(A\) only | \(A\cap B'\) | \(n(A)-n(A\cap B)\) |
| neither | \(A'\cap B'=(A\cup B)'\) | total \(-\ n(A\cup B)\) |
| \(A\) or not \(B\) | \(A\cup B'\) | everything except \(B\) only |
| \(B\) | \(B'\) | Total | |
|---|---|---|---|
| \(A\) | 6 | 11 | 17 |
| \(A'\) | 8 | 5 | 13 |
| Total | 14 | 16 | 30 |
How to fill in a Venn diagram from totals
- Start with the overlap (if it is not given, find it: \(n(A) + n(B) - n(A \cup B)\)).
- Fill the "only" regions: circle total minus the overlap.
- Fill the outside: grand total minus everything inside the circles.
- Check that all the regions add to the grand total.
How to find a probability
- Shade (or list) the regions that match the event.
- Add those regions for the numerator.
- Divide by the grand total (or, with probabilities in the regions, the total is \(1\)).
Writing the notation
- Translate each word: and \(\to \cap\), or \(\to \cup\), not \(\to\ '\).
- Shade one part at a time: for \(A \cap B'\), shade \(A\), then keep only the part outside \(B\).
- Count or add the probabilities in the shaded regions.
Total: \(11 + 6 + 8 + 5 = 30\).
| \((\text{i})\quad n(A \cup B)\) | \(=\) | \(11 + 6 + 8 = \boldsymbol{25}\) |
| \((\text{ii})\quad P(A \cap B)\) | \(=\) | \(\dfrac{6}{30} = \boldsymbol{\dfrac{1}{5}}\) |
(iii) \(B'\) is everything outside circle \(B\): \(A\) only and outside both.
| \(P(B')\) | \(=\) | \(\dfrac{11 + 5}{30} = \boldsymbol{\dfrac{8}{15}}\) |
(iv) Exactly one means \(A\) only or \(B\) only, not the overlap.
| \(P\) | \(=\) | \(\dfrac{11 + 8}{30} = \boldsymbol{\dfrac{19}{30}}\) |
(i) How many do both?
(ii) Find the probability that a student chosen at random plays an instrument only.
(iii) Find the probability that a student chosen at random does exactly one of the two.
(i) Students in at least one circle: \(90 - 14 = 76\). Adding the circles counts the overlap twice, so the extra is the overlap.
| \(\text{both}\) | \(=\) | \(48 + 41 - 76 = \boldsymbol{13}\) |
| \(\text{sport only}\) | \(=\) | \(48 - 13 = 35\) |
| \(\text{instrument only}\) | \(=\) | \(41 - 13 = 28\) |
| \((\text{ii})\quad P\) | \(=\) | \(\dfrac{28}{90} = \boldsymbol{\dfrac{14}{45}}\) |
(iii) Exactly one means sport only or instrument only, not the overlap.
| \(P(\text{exactly one})\) | \(=\) | \(\dfrac{35 + 28}{90} = \boldsymbol{\dfrac{7}{10}}\) |
(i) \(P(A \cap B)\)
(ii) \(P(A \cup B)\)
(iii) \(P(A' \cap B)\).
List each set, then find the numbers in both.
| \(A\) | \(=\) | \(\{4, 8, 12, 16, 20\}\) |
| \(B\) | \(=\) | \(\{1, 2, 4, 5, 10, 20\}\) |
| \(A \cap B\) | \(=\) | \(\{4, 20\}\) |
(i) Two numbers are in both sets.
| \(P(A \cap B)\) | \(=\) | \(\dfrac{2}{20} = \boldsymbol{\dfrac{1}{10}}\) |
(ii) \(A \cup B\) has \(5 + 6 - 2 = 9\) numbers (the \(2\) shared numbers are counted once).
| \(P(A \cup B)\) | \(=\) | \(\boldsymbol{\dfrac{9}{20}}\) |
(iii) \(A' \cap B\) means in \(B\) but not in \(A\): \(\{1, 2, 5, 10\}\).
| \(P(A' \cap B)\) | \(=\) | \(\dfrac{4}{20} = \boldsymbol{\dfrac{1}{5}}\) |
(i) \(P(A \cap B)\)
(ii) \(P(A \cap B')\)
(iii) \(P(A \cup B')\).
All four regions add to \(1\), so \(P(A \cup B) = 1 - 0.25 = 0.75\).
(i) Add \(P(A)\) and \(P(B)\), then take away \(P(A \cup B)\), because the overlap was counted twice.
| \(P(A \cap B)\) | \(=\) | \(0.55 + 0.4 - 0.75\) |
| \(\) | \(=\) | \(\boldsymbol{0.2}\) |
(ii) \(A \cap B'\) is \(A\) only.
| \(P(A \cap B')\) | \(=\) | \(0.55 - 0.2 = \boldsymbol{0.35}\) |
The regions are \(0.35\) (\(A\) only), \(0.2\) (both), \(0.4 - 0.2 = 0.2\) (\(B\) only) and \(0.25\) (neither). Check: they add to \(1\) ✓
(iii) \(A \cup B'\) is everything in \(A\) or outside \(B\): every region except \(B\) only.
| \(P(A \cup B')\) | \(=\) | \(1 - 0.2 = \boldsymbol{0.8}\) |
Common pitfalls
Frequently asked questions
What do the symbols intersection and union mean?
\(A \cap B\) (intersection) means \(A\) and \(B\), the overlap of the circles. \(A \cup B\) (union) means \(A\) or \(B\) or both, everything inside the two circles.
What does A' mean in a Venn diagram?
\(A'\) is the complement of \(A\): everything that is not in \(A\), which is all the regions outside circle \(A\).
How do you find the overlap of a Venn diagram?
Add the two circle totals and subtract the number in at least one circle. The extra amount is the overlap, because it was counted twice.
What are mutually exclusive events?
Events that cannot happen at the same time. Their circles do not overlap, so \(P(A \cup B) = P(A) + P(B)\).
How do you turn a Venn diagram into a two-way table?
Use \(A\) and \(A'\) as the rows and \(B\) and \(B'\) as the columns. The overlap goes in the \(A\), \(B\) cell, \(A\) only in \(A\), \(B'\), \(B\) only in \(A'\), \(B\), and the outside region in \(A'\), \(B'\). Then add the totals.