Statements and Open Sentences

The Big Idea

Compare these three pieces of mathematical language:

• \(3+5\)
• \(3+5=8\)
• \(x+5=8\)

The first is an expression. It represents a value but does not make a complete claim. The second is a statement. It makes a claim that is true. The third is an open sentence. Its truth depends on the value assigned to \(x\).

This small distinction leads directly to equation solving, inequalities, functions, logic, proof, set-builder notation, and mathematical modeling. Before we can reason about a sentence, we need to know whether it already has a truth value or is still waiting for information.

A Statement Makes a Definite Claim

A mathematical statement, also called a proposition, is a declarative sentence that is either true or false under the definitions and assumptions being used.

A false sentence is still a statement. “A triangle has four sides” is mathematically wrong, but it makes a clear claim whose truth value can be determined.

Truth Value Is Not the Same as Agreement

The truth value of a statement is whether it is true or false. We determine that value from definitions, assumptions, valid reasoning, calculation, or previously established results—not from a vote or personal preference.

A person may not yet know whether a statement is true. That lack of knowledge does not automatically turn the statement into an open sentence. For example, a long numerical claim may be difficult to check, but it is still a definite claim if all of its objects and conditions have been specified.

Mathematical truth also belongs to a context. The statement “the angles of a triangle sum to \(180^\circ\)” is true in Euclidean geometry. Other geometries use different assumptions and may give different angle sums. A careful statement includes or inherits the definitions and assumptions needed to interpret it.

Not Every Sentence Is a Statement

Questions, commands, and undefined judgments do not ordinarily have mathematical truth values.

We can often turn an informal sentence into a statement by adding precision. “This number is very large” might become “This integer is greater than one million.” Once the object and boundary are specified, the claim can receive a truth value.

An Open Sentence Waits for a Value

An open sentence contains one or more variables and becomes true or false only after those variables are assigned values or governed by quantifiers.

Consider

\[ x+5=8. \]

We cannot label this sentence simply true or false until we know what \(x\) represents. If \(x=3\), the sentence becomes \(3+5=8\), which is true. If \(x=4\), it becomes \(4+5=8\), which is false.

Other open sentences include:

• \(n\) is a prime number.
• \(y\lt12\).
• \(a^2+b^2=c^2\).
• The point \((x,y)\) lies on the circle \(x^2+y^2=25\).

Each sentence contains unresolved symbols. Its truth depends on their values and on the domain from which those values may be chosen.

Substitution Closes the Sentence

Substituting a value for every free variable turns an open sentence into a statement. Let

\[ P(x):\quad 2x+1\gt7. \]

The label \(P(x)\) gives the open sentence a convenient name. Now test a few values:

Substitution is therefore more than replacing letters with numbers. It lets us test which members of a domain make the open sentence true.

Open Sentences as Predicates

In logic, an open sentence is often called a predicate. A predicate describes a condition that an object may or may not satisfy.

The notation

\[ E(n):\quad n\text{ is even} \]

names a predicate \(E\) involving the variable \(n\). The statement \(E(12)\) is true, while \(E(9)\) is false.

A predicate behaves somewhat like a truth-valued function: input an allowed object, and the condition produces the truth value true or false. This viewpoint connects algebra, logic, computer programming, databases, and set theory.

The Domain Is Part of the Meaning

The domain, sometimes called the universe of discourse, is the collection of values allowed for the variables. The same open sentence can have a different solution set when the domain changes.

Consider

\[ x^2=4. \]

• Over the real numbers, the solutions are \(x=-2\) and \(x=2\).
• Over the integers, the solutions are still \(x=-2\) and \(x=2\).
• Over the positive integers, the only solution is \(x=2\).

Now consider

\[ x^2=-1. \]

This open sentence has no real solution, but it has the complex solutions \(x=i\) and \(x=-i\). Always ask which values are eligible before deciding which ones make the sentence true.

The Solution Set Collects the True Cases

The solution set or truth set of an open sentence contains every domain value that makes the sentence true.

For the open sentence

\[ x+5=8, \]

the solution set over the real numbers is

\[ S=\{3\}. \]

For

\[ x^2\lt9, \]

the solution set over the real numbers is the interval \((-3,3)\). Over the integers, the solution set is

\[ \{-2,-1,0,1,2\}. \]

Solving an equation or inequality means finding this entire set, not merely producing one value that happens to work.

Equations and Inequalities Are Often Open Sentences

An equation containing an unresolved variable is an open sentence because its truth depends on that variable. The same is true of an inequality.

The algebraic steps used in solving are truth-preserving transformations. We rewrite the open sentence without changing which values make it true.

Always True, Sometimes True, or Never True

Open sentences can be classified by how often they are true across a domain.

An identity written with a variable is often understood as carrying an implicit “for every permitted value.” The expression \(2(x+1)=2x+2\) may look open, but its intended claim is usually universal: it is true for every real \(x\).

Quantifiers Close Open Sentences

A quantifier tells us how many objects in a domain must satisfy a predicate. Two quantifiers appear throughout mathematics:

• Universal: “for every” or “for all,” written \(\forall\)
• Existential: “there exists” or “for at least one,” written \(\exists\)

Start with the open sentence \(x^2\ge0\). Over the real numbers, we can close it with a universal quantifier:

\[ \forall x\in\mathbb{R},\quad x^2\ge0. \]

This is a true statement: every real number has a nonnegative square.

We can also write

\[ \exists n\in\mathbb{Z}\quad\text{such that}\quad n^2=9. \]

This statement is true because at least one integer—indeed, \(3\) and \(-3\)—satisfies the condition.

Quantifiers do not assign a particular value. They turn an open condition into a complete claim about a domain.

Statements Can Be Combined

Simple statements can be joined to form compound statements. Words such as and, or, not, and if…then affect the truth conditions of the whole sentence.

• “\(12\) is even and \(12\) is divisible by 3” is true because both parts are true.
• “\(7\) is even or \(7\) is prime” is true because at least one part is true.
• “\(9\) is not a perfect square” is false because \(9=3^2\).
• “If an integer is divisible by 4, then it is even” is true.

Formal logic develops the precise rules for these connections. For now, the useful habit is to notice the smaller claims and the word that connects them.

Translate Between Words and Symbols

Translating an open sentence helps expose its objects, domain, and truth condition.

A symbolic sentence should preserve the strength of the words. “Every” is not interchangeable with “some,” and “less than” is not interchangeable with “less than or equal to.”

Context Can Supply Missing Information

A sentence that looks open in isolation may become a statement when earlier context has already assigned its variables. If a problem first says \(x=4\), then the later sentence \(x+1=5\) is a true statement within that context.

Context may also supply a domain, definitions, units, assumptions, or a diagram. “The point lies inside the circle” is incomplete if no point or circle has been identified. It becomes a definite statement once both objects are specified.

Good mathematical writing makes essential context easy to recover. Define variables, name domains, state assumptions, and avoid asking the reader to guess which objects a pronoun or symbol refers to.

A Friendly Sentence-Checking Routine

When you read mathematical language, ask:

1. Is this an expression, question, command, or declarative sentence?
2. If it is declarative, can it already be labeled true or false?
3. Are any variables still unresolved? If so, the sentence may be open.
4. What domain supplies the allowed values?
5. Would substitution turn the sentence into a definite statement?
6. Is there an explicit or implied quantifier? Look for every, all, some, or there exists.
7. What is the solution or truth set? Collect every permitted value that makes the sentence true.
8. Which assumptions and definitions determine the truth conditions?

This routine turns equation solving into a logical task: you are finding precisely when a mathematical claim becomes true.

Working Summary

A statement is a definite mathematical claim with a truth value. An open sentence contains unresolved variables and becomes a statement through substitution, assignment, or quantification.

The domain determines which values are allowed, and the solution set collects the values that make an open sentence true. Equations and inequalities are therefore not merely arrangements of symbols; they are truth conditions waiting to be analyzed.

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