Mathematical statements

A mathematical statement is a sentence that is either true or false, but not both, and not “undetermined.” This chapter focuses on what makes a sentence a mathematical statement, how statements are used in mathematics, and how they are typically written and read.

Because this chapter is part of “Mathematical Language,” we will treat statements as the basic “sentences” of that language. Logical symbols and connectives will be treated more fully in later chapters on logic; here we concentrate on recognizing and handling statements themselves.

What counts as a mathematical statement?

A mathematical statement (often just called a “statement” or “proposition”) is:

• A complete sentence,
• About some mathematical objects or relationships,
• That has a definite truth value: it is either true or false.

The key idea is definiteness. There is no room for “maybe,” “sometimes,” or “depends what you mean.” Once you understand all the terms and the context, a statement must come out either true or false.

Examples of mathematical statements (each is clearly true or clearly false):

• $2 + 3 = 5$ (true)
• $7$ is an even number. (false)
• $5 > 10$ (false)
• There exists a prime number greater than $100$. (true)
• For all real numbers $x$, $x^2 \ge 0$. (true)
• $1/2 = 0.5$ (true)

Each of these says something precise that we can, in principle, check.

Sentences that are not mathematical statements

Not every sentence is a mathematical statement. Some sentences are not statements because they are:

• Questions
• Commands
• Exclamations
• Incomplete
• Too vague to have a definite truth value

Examples:

• “What is $2+2$?” — a question, not a statement.
• “Solve the equation $x+3=5$.” — a command, not a statement.
• “Close the window.” — not something that can be true or false.
• “This number is big.” — vague, unless “big” is defined in some precise way.
• “$x+2=7$” — may or may not be a statement, depending on context (see below).

Mathematics mostly works with declarative sentences: sentences that declare something and therefore can be true or false.

Variables and context: open vs closed sentences

A sentence containing a variable can behave differently depending on how the variable is treated.

Consider:

• “$x+2=7$.”

On its own, this is not yet a statement in the strict sense, because you haven’t said what $x$ is allowed to be. It is like saying: “someone plus two equals seven.” It is an open sentence: its truth depends on the value of $x$.

However, in an appropriate context, this can be turned into a statement. For example:

• “There exists a real number $x$ such that $x+2=7$.”

Now this is a statement: it can be checked and is true.

Or:

• “For all integers $x$, $x+2=7$.”

This is also a statement: it can be checked and is false.

So:

• An open sentence contains variables whose possible values are not specified; it does not yet have a definite truth value.
• A closed sentence has enough information (for example, by specifying what values the variables can take, or by using words like “for all” or “there exists”) so that it does have a definite truth value. Closed sentences are mathematical statements.

Later chapters on variables, and on logic, will give more tools for handling open and closed sentences; here the important point is to notice which kind you are dealing with.

Types of mathematical statements

Even at a beginner level, it is useful to recognize some common kinds of mathematical statements. They show up everywhere, from arithmetic to advanced topics.

Numerical statements

These are about specific numbers and their relationships:

• $8+5=13$.
• $9$ is a multiple of $3$.
• $17$ is a prime number.
• $3/4 < 2$.

Such statements are often the first ones you see in school mathematics.

Equality and inequality statements

These use equality ($=$) and inequality symbols ($<,>,\le,\ge$):

• $4+1 = 5$.
• $7 \neq 9$.
• $-3 < 0$.
• $2x+1 \ge 5$ for all $x \ge 2$.

The last example is again a statement because it explicitly says “for all $x \ge 2$,” which closes the sentence.

Existence statements

These claim that at least one object with a certain property exists.

They usually have the form:

• “There exists [some kind of object] such that [property].”

Examples:

• There exists an integer $n$ such that $n^2 = 25$.
• There exists a rational number between $1$ and $2$.

In more symbolic form (explored later in logic), this kind of statement is often written with an existential quantifier. For now, focus on the words “there exists” as a clue.

Universal statements

These claim that every object of a certain kind has a certain property.

They usually have the form:

• “For all [objects of a certain kind], [property].”
• Or: “Every [object of a certain kind] has [property].”

Examples:

• For all natural numbers $n$, $n+0 = n$.
• Every even number is divisible by $2$.

The phrases “for all,” “every,” and “any” are signals that you are dealing with a universal statement.

Conditional (if–then) statements

These relate two statements in a cause–effect, or condition–consequence way.

They have the form:

• “If [condition], then [conclusion].”

Examples:

• If a number is even, then it is divisible by $2$.
• If $x>2$, then $x^2>4$.

Each part (“a number is even,” “it is divisible by $2$”) is itself a statement. The whole “if–then” is another statement that says something about how the two are connected.

Conditional statements are central in proofs and reasoning; their logical behavior is treated at length in later chapters on logic.

Definitions as statements

Mathematics often introduces definitions that look like statements:

• A number is even if it is divisible by $2$.
• A prime number is a natural number greater than $1$ that has no positive divisors other than $1$ and itself.

Definitions do not claim that something is true or false in the same way a fact does; instead, they explain how a word is to be used. Still, they are written in the same declarative style as ordinary statements, so they are part of the mathematical language that uses statements as building blocks.

Truth values: true, false, and why it matters

Each mathematical statement has a truth value: either true or false.

Deciding the truth value of a statement is a basic mathematical activity:

• To show a statement is true, you give a reason (a proof, or a calculation, or an argument).
• To show a statement is false, it often suffices to give an example where it fails.

For instance:

• Statement: “Every odd number is prime.”
• Counterexample: $9$ is odd but not prime.
• Conclusion: The statement is false.

A counterexample is a single example that shows a universal statement is not true. Learning to look for counterexamples is an important skill when dealing with mathematical statements.

Notice that even a false statement is still a valid statement. Being a statement does not require being true; it only requires having a definite truth value.

Why precise statements are important

Mathematics aims to be precise and unambiguous. Vague sentences lead to confusion, because different people might interpret them differently, and then they might not agree on what is true or false.

Compare:

• Vague: “Big numbers are hard to work with.”
• Precise: “For any fixed positive integer $k$, there are only finitely many integers with exactly $k$ digits.”

The first is not clearly mathematical: “big” and “hard” are not defined. The second is precise and has a definite truth value (it is true).

When you write or read mathematics, it helps to:

• Prefer clear quantifiers like “for all,” “there exists,” “exactly one.”
• Avoid phrases like “a lot,” “usually,” “small,” “big,” unless they are made precise.
• Make sure each sentence you rely on is a statement: it can be judged true or false.

How mathematical statements are used

Mathematical statements serve several roles:

• Assumptions or axioms: starting points accepted without proof within a system.
• Theorems or propositions: statements that have been proved.
• Lemmas and corollaries: helper statements that support or follow from other results.
• Conjectures: statements believed to be true but not yet proved or disproved.

All of these are statements in the logical sense; they differ in how we relate to their truth:

• An axiom is taken as true in the context.
• A theorem is known to be true because it has a proof.
• A conjecture is not yet known to be true or false, even though it has to be one or the other.

A famous example:

• “There are infinitely many prime numbers.” — This is a true statement (a theorem).
• “There are infinitely many twin primes (pairs of primes that differ by 2).” — This is a conjecture: it is a statement, but its truth value is currently unknown to mathematicians.

Recognizing mathematical statements: practice patterns

When reading or writing mathematics, it is helpful to practice classifying sentences. Here are some guiding questions:

1. Is this sentence declarative? (Not a question or a command.)
2. Does it talk about mathematical objects (numbers, sets, functions, etc.)?
3. Once all terms are defined and the context is fixed, can it be judged true or false?
4. If it contains variables, is it clear what values they can take, and is it clear whether “for all” or “there exists” is intended?

For each of the following, think about whether it is a mathematical statement:

• “$3 \cdot 4 = 12$.”
• “Find all $x$ such that $x^2 = 9$.”
• “There exists a real number $x$ such that $x^2 = -1$.”
• “Some numbers are very large.”
• “For every integer $n$, $n^2 + n$ is even.”

Only those that can be assigned a definite truth value, once their terms are understood, count as mathematical statements.

From ordinary language to mathematical statements

Many problems in mathematics start as everyday-language sentences and then get translated into precise mathematical statements.

Examples of such translations might look like:

• Everyday: “One number is five more than another, and their sum is $19$.”
• Mathematical form: “There exist integers $x$ and $y$ such that $x = y + 5$ and $x + y = 19$.”
• Everyday: “No even number greater than $2$ is prime.”
• Mathematical form: “For every even integer $n>2$, $n$ is not prime.”

Later chapters, especially those on algebra and logic, will focus on this process of translation. For now, it is enough to recognize that the goal of translation is to arrive at clear mathematical statements with definite truth values.

Summary

In this chapter, the focus is on recognizing and understanding mathematical statements as the basic sentences of mathematical language:

• A mathematical statement is a declarative sentence that is either true or false.
• Questions, commands, and vague sentences are not mathematical statements.
• Sentences with variables are not automatically statements; they become statements when the variables and quantifiers (like “for all,” “there exists”) are made clear.
• Common types of statements include numerical, equality/inequality, existence, universal, and conditional statements.
• Precision and definiteness are essential: the goal is to avoid ambiguity so that truth or falsity is well defined.
• Statements are the building blocks from which definitions, proofs, theorems, and conjectures are constructed.

Understanding what counts as a mathematical statement prepares you to work with logical connectives, quantifiers, and proofs in later chapters.