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Statements and truth · Introduction to Analysis | Math Infinitum

Intuition

A mathematical statement is a sentence that is either true or false, with nothing in between. Questions, instructions and opinions are not statements. Neither is a sentence holding a variable nobody has fixed: it has no truth value until the variable is given a value.

Think of a light switch, which is either on or off. A statement is a sentence wired to such a switch. "Close the door" has no switch to read at all, and "it is bigger than three" has a switch nobody has set.

Statements and open sentences

Everything in this chapter is built from sentences that carry a truth value. We write P, Q and R for statements. A sentence with a free variable is an open sentence, written P(x); it turns into a statement once the variable is fixed.

\begin{aligned}&P \text{ is a statement} \iff\\ &P \text{ is true or } P \text{ is false, never both}\end{aligned}

Reading the notation

$2 + 2 = 4$ is a statement, and it is true.
$2 + 2 = 5$ is a statement, and it is false. Being false does not stop a sentence being a statement.
$x > 3$ is an open sentence. Fix and it becomes a statement, and a true one.

Applications

Practice

Ask whether it can be judged

$7 > 3$ is a statement. "Is $7 > 3$ ?" is not.

The first asserts something we can check. The second only asks, and a question has nothing to judge.

Try it

Which of these is a mathematical statement?

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A false statement is still a statement

3 \in \{1,2\}

This is false, and that is precisely what makes it a statement: we could decide it, and the decision came out false.

Try it

Is $10 < 4$ a statement?

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Fixing the variable

Let $P(n)$ be $n > 4$ . Then $P(7)$ is true and $P(2)$ is false.

The same open sentence yields a true statement for one value and a false one for another.

Try it

Let $P(n)$ be $n^2 = 9$ . Which value makes $P(n)$ a true statement?

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Statements about sets

Let $A = \{1,2\}$ . Then $2 \in A$ is true and $5 \in A$ is false.

Membership always yields a statement: the object either is listed or it is not.

Try it

Let $A = \{1,2\}$ and $B = \{1,2,3\}$ . Which sentence is a false statement?

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Try it

Can a single mathematical statement be both true and false at once?

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Try it

How many of these four sentences are statements as they stand: " $7$ is prime"; " $x+1=4$ "; "Is $3$ odd?"; " $0=1$ "?

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Try it

The sentence " $n$ is even" is a statement.

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Two Statements, One Truth Value

Two statements are equivalent when they have the same truth value in every case — not when they look alike.

P\iff Q

Try it

Which pair of statements is equivalent?

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What You Learned

A statement has exactly one truth value: true or false, never both and never neither.
A question, an instruction and an opinion are not statements.
A sentence with a free variable is a predicate, and becomes a statement when the variable is fixed or quantified.
Being false does not stop a sentence being a statement.

Final checkpoint

Try it

Let $S=\varnothing$ . Which of these is a true statement?

Submit an answer to see feedback.

Completion

Lesson complete

Great work! You now know how to:

Tell a statement from a question, an instruction or a predicate
Fix or quantify a free variable to make a statement
Judge a vacuous universal claim correctly