Introduction to Mathematical Statements

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Section 1.1 Introduction to Mathematical Statements

Guiding Questions.

In this section, we’ll seek to answer the questions:

What is a statement?
What are some examples and non-examples of statements?
What is an atomic statement?

In order to do mathematics, we need an idea of what sorts of questions we can explore. We will focus our energies on statements. This implicitly limits the domain of mathematics, but in doing so we gain power in that domain.

Definition 1.1.1.

A statement is a declarative sentence that is either true or false.

Activity 1.1.2.

Determine whether the following are statements. If so, explain why, and determine their truth value if you are able. If not, explain why not.

Quadratic equations have at most two real solutions.
There exists a function that is differentiable at 0 but not continuous at 0.
The moon is made entirely of cheese.
What time is it?
The sum of the first \(n\) positive integers is \(\frac{n(n+1)}{2}\text{.}\)
That is beautiful!
A hexagon has six sides, and two distinct lines in the plane are either parallel or meet in exactly one point.
The Green Bay Packers are the worst football team.
\(\displaystyle 3+4=7.\)
This statement is false.

Activity 1.1.3.

Give at least one example and one non-example of a statement.

Most of the statements we’ve explored so far might be considered atomic in the sense that they cannot be broken down into combinations of simpler statements. In Section 1.2, we will explore ways of building more complex statements out of simpler ones via logical connectives.