Set Notation & Venn Diagrams

Set Notation Basics

A set is a collection of distinct objects, called elements or members. Sets are usually written using curly braces { } with the elements listed inside, separated by commas.

For example, the set of vowels in the English alphabet is written as {a, e, i, o, u}.

We use the symbol ∈ to mean "is an element of" or "belongs to". For example, a ∈ {a, e, i, o, u} means "a is an element of the set of vowels".

If an element is not in a set, we write ∉. For example, b ∉ {a, e, i, o, u} means "b is not an element of the set of vowels".

Sets can be finite or infinite, but at IGCSE level we mainly deal with finite sets.

For instance, the set of prime numbers less than 10 is {2, 3, 5, 7}.

Common Set Operations

Sets can be combined or compared using operations. The most common are union, intersection, and difference.

Union (∪)

The union of two sets A and B, written A ∪ B, is the set of all elements that are in A or B (or both).

For example, if A = {1, 2, 3} and B = {3, 4, 5}, then

$$A \cup B = \{1, 2, 3, 4, 5\}$$

Intersection (∩)

The intersection of two sets A and B, written A ∩ B, is the set of all elements that are in both A and B.

Using the same sets as above:

$$A \cap B = \{3\}$$

Difference (−)

The difference between two sets A and B, written A - B, is the set of elements that are in A but not in B.

For example:

$$A - B = \{1, 2\}$$

Note that B - A = \{4, 5\}.

The complement of a set A (usually written as A' or A^c) means all elements not in A but in the universal set U. The universal set contains all elements under consideration.

For example, if U = \{1, 2, 3, 4, 5, 6\} and A = \{1, 2, 3\}, then

$$A' = U - A = \{4, 5, 6\}$$

These operations are fundamental when working with sets and Venn diagrams.

For instance, if U = \{a, b, c, d, e\}, A = \{a, b, c\}, and B = \{b, d\}, then

$$A \cup B = \{a, b, c, d\}, \quad A \cap B = \{b\}, \quad A - B = \{a, c\}, \quad B - A = \{d\}$$

Venn Diagrams

Venn diagrams are visual tools to represent sets and their relationships. Each set is shown as a circle within a rectangle representing the universal set.

For two sets, two overlapping circles are drawn. The overlapping region represents the intersection.

For three sets, three overlapping circles are drawn so all possible intersections can be shown.

Regions in the diagram can be shaded to represent unions, intersections, or differences of sets.

For example, shading the entire area covered by two circles shows the union A ∪ B. Shading only the overlapping part shows the intersection A ∩ B.

Venn diagrams help to solve problems involving sets by making relationships clear.

Using Set Notation with Venn Diagrams

Set operations can be represented visually with Venn diagrams. Understanding how to interpret shaded regions and write the corresponding set notation is essential.

For example, shading all parts of two circles except their intersection represents the union minus the intersection, which can be written as (A ∪ B) - (A ∩ B).

You can also use Venn diagrams to find unknown values or verify set relationships.

For instance, if you know the number of elements in each region, you can calculate the total number of elements in unions or intersections.

Example: If n(A) = 5, n(B) = 7, and n(A ∩ B) = 3, then the number of elements in A ∪ B is

$$n(A \cup B) = n(A) + n(B) - n(A \cap B) = 5 + 7 - 3 = 9$$

For instance, if A = \{1, 2, 3\} and B = \{3, 4, 5\}, then $A \cup B = \{1, 2, 3, 4, 5\}$.