Sets and Venn diagrams

What is a set

A set is a collection of items, called elements, written inside curly brackets: $A = \{2, 4, 6, 8\}$. A set can also be described in words, such as B = {multiples of 3} or C = {prime numbers}.

• $x \in A$ means $x$ is an element of set $A$ — here $4 \in A$.
• The universal set $E$ is everything being considered in the question. In a Venn diagram it is the rectangle drawn around the circles.
• $A \subset B$ means $A$ is a subset of $B$: every element of $A$ is also in $B$.

Union and intersection

• The union $A \cup B$ contains every element that is in $A$, in $B$, or in both — the two circles combined.
• The intersection $A \cap B$ contains only the elements that are in both sets — the overlap of the circles.

If $A = \{1, 3, 5, 7\}$ and $B = \{3, 4, 5\}$, then $A \cup B = \{1, 3, 4, 5, 7\}$ and $A \cap B = \{3, 5\}$. In a union each element is written once, even if it belongs to both sets.

Complement and counting

• The complement $\overline{A}$ is every element of $E$ that is not in $A$. If $E$ is the whole numbers from 1 to 10 and $A$ = {even numbers}, then $\overline{A} = \{1, 3, 5, 7, 9\}$.
• $|A|$ means the number of elements in $A$. Here $|A| = 5$.

Symbols combine — read them from the inside out. $\overline{A \cup B}$ is everything outside both circles, and $|A \cap B|$ counts the elements in the overlap.

Venn diagram word problems

A typical question: in a class of 28 students, 15 play hockey, 14 play cricket and 6 play both.

1. Fill in the overlap first: 6 play both.
2. Subtract to find each “only” region: hockey only is $15 - 6 = 9$; cricket only is $14 - 6 = 8$.
3. Whatever remains sits outside the circles: $28 - (9 + 6 + 8) = 5$ play neither sport.

Once the diagram is complete you can answer anything: “exactly one sport” is $9 + 8 = 17$, “at least one” is $23$.