Skip to main content

Beginning Activity Beginning Activity 2: Venn Diagrams for Two Sets
A4US

In Beginning Activity 1, we worked with verbal and symbolic definitions of set operations. However, it is also helpful to have a visual representation of sets. Venn diagrams are used to represent sets by circles (or some other closed geometric shape) drawn inside a rectangle. The points inside the rectangle represent the universal set \(U\text{,}\) and the elements of a set are represented by the points inside the circle that represents the set. For example, FigureĀ 5.1 is a Venn diagram showing two sets.

Venn Diagram with two circles A and B inside the universal set U. A overlaps B creating 3 inner-circle regions. The leftmost is labelled 1, the center 2, and the right 3. The area outside the circles but in U is labelled 4.
Figure 5.1. Venn Diagram for Two Sets

In FigureĀ 5.1, the elements of \(A\) are represented by the points inside the left circle, and the elements of \(B\) are represented by the points inside the right circle. The four distinct regions in the diagram are numbered for reference purposes only. (The numbers do not represent elements in a set.) The following table describes the four regions in the diagram.

Region Elements of \(U\) Set
1 In \(A\) and not in \(B\) \(A - B\)
2 In \(A\) and in \(B\) \(A \cap B\)
3 In \(B\) and not in \(A\) \(B - A\)
4 Not in \(A\) and not in \(B\) \(A^c \cap B^c\)
We can use these regions to represent other sets. For example, the set \(A \cup B\) is represented by regions 1, 2, and 3 or the shaded region in FigureĀ 5.2.
Venn Diagram with two circles A and B inside the universal set U. A overlaps B creating 3 inner-circle regions. The leftmost is labelled 1, the center 2, and the right 3. The area outside the circles but in U is labelled 4. Areas 1, 2, and 3 shaded.
Figure 5.2. Venn Diagram for \(A \cup B\)

Let \(A\) and \(B\) be subsets of a universal set \(U\text{.}\) For each of the following, draw a Venn diagram for two sets and shade the region that represent the specified set. In addition, describe the set using set builder notation.

1.

\(A^c\)

2.

\(B^c\)

3.

\(A^c \cup B\)

4.

\(A^c \cup B^c\)

5.

\(\left( A \cap B \right)^c\)

6.

\(\left( A \cup B \right) - \left( A \cap B \right)\)