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Beginning Activity Beginning Activity 1: Statements
A4US

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 ā€œThe sky is beautifulā€ is not a statement since whether the sentence is true or not is a matter of opinion. A question such as ā€œIs it raining?ā€ 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.

1.

\(3 \cdot 4 + 7 = 19\text{.}\)

2.

\(3 \cdot 5 + 7 = 19\text{.}\)

3.

\(3x + 7 = 19\text{.}\)

4.

There exists an integer \(x\) such that \(3x + 7 = 19\text{.}\)

5.

The derivative of \(f(x) = \sin x\) is \(f'(x) = \cos x\text{.}\)

6.

Does the equation \(3x^2 - 5x - 7 = 0\) have two real number solutions?