The algebra of events

What you will be able to do

• build the events "$A$ or $B$", "$A$ and $B$", "not $A$" and "$A$ but not $B$";
• write "exactly one", "neither" and "at least one" as sets;
• count with $n(A \cup B) = n(A) + n(B) - n(A \cap B)$ and use De Morgan's laws.

What the words mean

In a school, some students play chess and some play table tennis. A teacher asks four questions. Who plays chess or table tennis? Who plays both? Who plays neither? Who plays chess but not table tennis? Each question is a new event made from two old ones.

Let $A$ and $B$ be events of the same sample space $S$.

• $A \cup B$ (union) is the event "$A$ or $B$". It happens when at least one of them happens, and it holds the outcomes that are in $A$, in $B$ or in both.
• $A \cap B$ (intersection) is the event "$A$ and $B$". It happens when both happen.
• $A'$ (complement, also written $\bar{A}$ or $A^c$) is the event "not $A$". It holds every outcome of $S$ that is not in $A$.
• $A - B$ (difference) is the event "$A$ but not $B$". It is the same as $A \cap B'$. Some books write $A \setminus B$ for this event.

Longer sentences

• Exactly one of $A$ and $B$ is $(A - B) \cup (B - A)$. It is $A \cup B$ without $A \cap B$.
• Neither $A$ nor $B$ is $A' \cap B'$. It is the complement of the union.
• At least one is the union. Both is the intersection.
• The three parts $A - B$, $A \cap B$ and $B - A$ are mutually exclusive, and together they make $A \cup B$.

De Morgan's laws

In words, "not (A or B)" means "not A and not B". "Not (A and B)" means "not A or not B". A Venn diagram shows both at once. A student is outside the union exactly when the student is outside $A$ and outside $B$.

Counting the union

When outcomes are equally likely, counting is all we need. Add $n(A)$ and $n(B)$. The outcomes in the overlap have been counted twice, so take away $n(A \cap B)$ once:

For mutually exclusive events the overlap is empty, and the rule is just $n(A) + n(B)$.

Three events

For three events $A$, $B$ and $C$ there are eight regions. The picture shows them. An outcome may belong to only one of the events, to two, to all three, or to none.

The count of the union becomes

The idea is the same as for two events: add, take away the overlaps once, then give back the middle part that was taken away too many times. The same rule works for probabilities, as you will see in the next course.

A worked example

In a class of $40$ students, $25$ study French, $18$ study German and $8$ study both. Let $F$ and $G$ be the events that a student, picked from the class, studies French or German.

1. At least one language. $n(F \cup G) = 25 + 18 - 8 = 35$.
2. Neither language. $n((F \cup G)') = 40 - 35 = 5$.
3. French only. $n(F - G) = 25 - 8 = 17$. German only is $18 - 8 = 10$.
4. Exactly one language. $17 + 10 = 27$. Check: $35 - 8 = 27$. The two ways agree.

Practise with new numbers

Try it in Python

Python sets have the same operations as events. The program uses one die, with $A$ for "at most $4$" and $B$ for "even". It prints each new event and checks De Morgan's laws.

Common mistakes

• Reading "or" as "exactly one". In mathematics, "$A$ or $B$" includes the case where both happen.
• Forgetting to take away the overlap. $n(A) + n(B)$ counts the overlap twice.
• Confusing $A - B$ with $B - A$. The order matters.
• Taking the complement of the wrong thing. $(A \cup B)'$ is not $A' \cup B'$. It is $A' \cap B'$.

Remember these

• Union is "or", intersection is "and", complement is "not", and $A - B = A \cap B'$.
• Exactly one of $A$ and $B$ is $(A \cup B) - (A \cap B)$. Neither is $A' \cap B'$.
• $n(A \cup B) = n(A) + n(B) - n(A \cap B)$.
• De Morgan: $(A \cup B)' = A' \cap B'$ and $(A \cap B)' = A' \cup B'$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. In a group of $30$ people, $16$ like tea, $14$ like coffee and $6$ like both. How many like at least one of them?

2. Using the numbers of question 1, how many like exactly one of the drinks?

3. A die is thrown. $A = \{1, 2, 3\}$ and $B = \{3, 4\}$. Write $A \cup B$, $A \cap B$ and $A - B$.

4. Write "neither $A$ nor $B$" in two ways using the operations.

5. Is $A - B$ always the same as $B - A$?

References

• NCERT, Mathematics for Class XI (earlier edition), Probability, section on the algebra of events.
• OpenStax, Introductory Statistics, Chapter 3, sections on terminology and on Venn diagrams.