3.1 Euler diagrams

In this section, we introduce a way of representing statements. These will be most useful later in the chapter when we discuss conditional statements, and when we discuss proof techniques.

In an Euler¹¹ 1 Leonard Euler (1707–1783 CE) was a prolific Swiss mathematician, whose surname is pronounced ‘OY-ler’. diagram, we represent a statement with a closed shape. For example, we can represent a statement $P$ as shown in Figure 3.1.

Figure 3.1: Representation of a statement

P

using an Euler diagram

We indicate that statement $P$ is true by putting a dot inside the shape representing statement $P$ . We indicate that $P$ is false by putting a dot outside the shape.

(a) Statement

P

is true

(b) Statement

P

is false

Figure 3.2: Representing whether a statement is true or false

The rectangle enclosing the shape represents the ‘universe’ in which the statement exists. When placing a dot, it must be inside the rectangle. A dot must be either inside the shape representing $P$ or outside it, indicating that $P$ must either be true or false, but not both.