Mathematical Reasoning: Writing and Proof (PreTeXt Edition)

Much of our work in mathematics deals with statements. In mathematics, a statement is a declarative sentence that must have a definite truth value, either true or false but not both. A statement is sometimes called a proposition. The key is that there must be no ambiguity. To be a statement, a sentence must be true or false, and it cannot be both. So a sentence such as “The sky is beautiful” is not a statement since whether the sentence is true or not is a matter of opinion. A question such as “Is it raining?” is not a statement because it is a question and is not declaring or asserting that something is true.

Some sentences that are mathematical in nature often are not statements because we may not know precisely what a variable represents. For example, the equation $2x+5=10$ is not a statement since we do not know what $x$ represents. If we substitute a specific value for $x$ (such as $x=3$), then the resulting equation, $2\cdot 3+5=10$ is a statement (which is a false statement).

Which of the following sentences are statements? Do not worry about determining the truth value of those that are statements; just determine whether each sentence is a statement or not.