Venn Diagrams and Set Notation

A Venn diagram sorts things into groups called sets. Each set is drawn as a circle, and a rectangle around the circles holds everything in the question. Where two circles overlap, the things inside belong to both sets.

In the exam you are usually given some sets of numbers and asked to complete the Venn diagram for this information, for 3 or 4 marks. A second part then asks for a probability, such as the probability that a number chosen at random is in the set A \cap B, for 2 marks.

You may instead be given a finished Venn diagram and asked to write down the numbers in a set such as P'. These questions come late in the paper, and many students lose marks because they don't know the symbols. Learn the three symbols on this page and the marks are within reach.

A set is a list of things, written inside curly brackets. A = \{2, 4, 6, 8\} means that set A contains the numbers 2, 4, 6 and 8. Each thing in a set is called a member of the set.

The universal set, written \mathcal{E}, holds everything in the question. In a Venn diagram it is the rectangle. \mathcal{E} = \{1, 2, 3, \ldots, 10\} means every whole number from 1 to 10, so all ten numbers must appear somewhere in the rectangle. Some books write \xi instead of \mathcal{E}

Two overlapping circles split the rectangle into four regions. Every member of \mathcal{E} is written exactly once, in the one region where it belongs.

A number in the overlap belongs to both sets, so it is not written again anywhere else in the circles. A number in neither set still goes in the rectangle, outside the circles.

A set is often given by a rule, such as A = \{\text{square numbers}\}. Before you draw anything, list the members that are in \mathcal{E}. If \mathcal{E} = \{1, 2, 3, \ldots, 20\}, the square numbers are 1, 4, 9 and 16

Watch the number 1. It is a square number, because 1 \times 1 = 1, and it is a factor of every whole number. It is not a prime number.

Only list numbers that are in \mathcal{E}. If \mathcal{E} stops at 12, the multiples of 5 are just 5 and 10

A \cap B means the members that are in A and in B. It is the overlap.

A \cup B means the members that are in A or B or both. It is everything inside the circles. The symbol \cup looks like a U.

A' means not in A: every member of \mathcal{E} outside circle A. That includes the numbers in B only and the numbers outside both circles.

When a number is chosen at random from \mathcal{E}, every member of \mathcal{E} is equally likely to be chosen. Count the members in the set you want, then count all the members of \mathcal{E}

P(\text{in the set}) = \frac{\text{how many are in the set}}{\text{how many are in } \mathcal{E}}

The bottom of the fraction counts everything in the rectangle, including the numbers outside the circles. Give the probability as a fraction, such as \frac{4}{10}. You may simplify it, but you don't have to.

Three overlapping circles make eight regions: the middle, where all three circles overlap, three regions where just two circles overlap, three regions for one set only, and the outside.

Fill them in from the middle outwards. First the numbers in all three sets, then the numbers in exactly two sets, then the numbers in one set only, and last the numbers in none of the sets.

Each number still goes in exactly one region. A number in all three sets is written once, in the middle.

• Writing a number in more than one region. Each member of \mathcal{E} goes in exactly one region, so a number in the overlap is not written again in the circles.
• Leaving out the numbers outside the circles. Every member of \mathcal{E} must appear, so write the leftover numbers in the rectangle, outside the circles.
• Putting 1 in the wrong place. 1 is a square number and a factor of every number, but it is not prime.
• Giving A when the question asks for A'. A' means everything not in A, including the numbers outside both circles.
• Mixing up \cap and \cup. A \cap B is the overlap only; A \cup B is everything inside either circle.
• Dividing by the wrong total. For a probability, divide by the number of members of \mathcal{E}, counting the ones outside the circles too.