Precision Begins Before the Proof
Before proving a claim, a reader must be able to determine exactly what the claim says. A sentence can sound plausible while leaving important questions unanswered: Which numbers are being considered? Does the sentence concern every number or just one? Does a stated condition need to imply a conclusion, or are the two properties meant to hold together? Mathematical precision makes these choices explicit.
The previous tutorial, How to Debug a Proof, emphasized recording a claim's domain, hypotheses, quantifiers, and conclusion before tracing its proof. This tutorial focuses on writing those components clearly in the first place. A precise sentence is not necessarily long or formal-looking. It is one whose intended mathematical meaning does not depend on a reader guessing what was meant.
A useful way to draft a claim is to identify its ingredients before choosing the final wording. Name the objects, state the domain from which they are drawn, specify how many objects are involved, and make the relationship between their properties explicit. If a sentence includes more than one variable, also check which quantifier applies to each variable.
Make the Objects and Their Domains Visible
A variable is a placeholder for an object. Its domain tells the reader which objects are allowed to fill that placeholder. The sentence “Let \(n\) be an integer” specifies a different domain from “Let \(x\) be a real number.” If the claim relies on a property of integers, leaving the domain unstated can make it unclear whether the claim is intended for integers, real numbers, or some other collection.
For a general claim, phrases such as “for every real number \(x\)” or “there exists an integer \(n\)” state both the quantity of objects under consideration and their domain. In symbols, these can be written \(\forall x\in\mathbb R\) and \(\exists n\in\mathbb Z\). The variable letter itself carries no special domain: \(x\) does not mean “real number” unless the sentence says so or the context has already fixed that convention.
Avoid relying on a pronoun when it is unclear which object the pronoun refers to. For instance, “For each real number \(x\), choose a real number \(y\) larger than it” is understandable, but “it” can be replaced with the variable itself: “For each real number \(x\), there is a real number \(y\) such that \(y>x\).” The latter sentence makes the relationship explicit and is straightforward to translate into symbols.
Worked Example: Replacing an Unspecified Number
Consider the sentence, “A number has a larger number.” It does not tell us whether the claim concerns one number or every number, and it does not specify a domain. A precise version of one likely intention is: “For every real number \(x\), there exists a real number \(y\) such that \(y>x\).”
In symbols, the statement is $$ \forall x\in\mathbb R,\ \exists y\in\mathbb R,\quad y>x. $$ To verify it, let \(x\in\mathbb R\) be arbitrary. Set \(y=x+1\). Since the real numbers are closed under addition, \(y\in\mathbb R\), and since \(1>0\), we have \(y=x+1>x\). Thus every real number has a larger real number.
The sentence “There exists a real number \(x\) that has a larger real number” would be weaker: it asks for one successful input rather than every input. For example, \(x=0\) and \(y=1\) satisfy \(y>x\). These two English sentences make different claims, even though both mention a number and a larger number.
Let the Words Match the Logical Structure
Words such as “every,” “some,” “if,” “and,” and “or” are not interchangeable. They tell the reader how conditions fit together. If a sentence says “if \(P\), then \(Q\),” it asserts that whenever the condition \(P\) holds, the conclusion \(Q\) follows. It does not assert that \(P\) and \(Q\) are both true for every object. When writing such a claim, clearly mark the condition and the conclusion.
The tutorials on logical connectives and on necessary and sufficient conditions established the meanings of these logical relationships. Here the practical point is to use language that expresses the relationship you intend. “If \(x\) is divisible by \(8\), then \(x\) is divisible by \(4\)” has a stated condition and a stated consequence. “\(x\) is divisible by \(8\) and \(4\)” instead joins two properties; it does not say that the first property guarantees the second.
A sentence using “only if” deserves particular care. The phrase does not have the same order as the phrase “if.” The tutorials on implication and necessary and sufficient conditions give the precise relationship. When in doubt, write the conditional with its hypothesis and conclusion, or express the two directions separately if the intended claim is an equivalence. Do not assume the word order alone will be interpreted as intended.
Worked Example: Separating a Hypothesis From a Conclusion
Write a precise version of: “A number divisible by \(8\) is divisible by \(4\).” Because divisibility is a property of integers, state the domain and the implication explicitly: “For every integer \(n\), if \(n\) is divisible by \(8\), then \(n\) is divisible by \(4\).”
Using the notation \(8\mid n\) for “\(n\) is divisible by \(8\),” and \(4\mid n\) for “\(n\) is divisible by \(4\),” this becomes $$ \forall n\in\mathbb Z,\quad (8\mid n\Longrightarrow 4\mid n). $$ To prove it, let \(n\in\mathbb Z\) and suppose \(8\mid n\). By the definition of divisibility, there is an integer \(k\) such that \(n=8k\). Since \(k\in\mathbb Z\), the number \(m=2k\) is an integer. We have \(n=8k=4(2k)=4m\), so \(4\mid n\). This proves the stated conditional.
Notice how the wording determines the proof's opening. “For every integer \(n\)” tells us to begin with an arbitrary integer, while “if \(8\mid n\)” tells us what to assume before deriving the conclusion \(4\mid n\). If the sentence omitted the condition, it would assert that every integer is divisible by \(4\), which is false; \(n=1\) is a counterexample.
Quantifier Order and Scope
When a sentence contains more than one quantifier, their order can change its meaning. The tutorial on understanding quantifier order established that the order of unlike quantifiers generally cannot be exchanged. In writing, this means that “for every \(x\), there is a \(y\)” and “there is a \(y\) that works for every \(x\)” must not be treated as stylistic variants.
A practical test is to ask whether the object chosen later may depend on the object introduced earlier. In “for every \(x\), there exists a \(y\),” the choice of \(y\) can depend on \(x\). In “there exists a \(y\) such that for every \(x\),” one fixed \(y\) must work for all \(x\). If that dependence matters, write the variables and quantifiers explicitly rather than relying on a compressed sentence.
Worked Example: Distinguishing Two Quantifier Orders
Compare the claims “For every real number \(y\), there exists a real number \(x\) with \(x<y\)” and “There exists a real number \(x\) such that \(x<y\) for every real number \(y\).” Their symbolic forms are, respectively, $$ \forall y\in\mathbb R,\ \exists x\in\mathbb R,\quad x<y $$ and $$ \exists x\in\mathbb R,\ \forall y\in\mathbb R,\quad x<y. $$
The first statement is true. Let \(y\in\mathbb R\) be arbitrary and choose \(x=y-1\). Then \(x\in\mathbb R\) and \(x=y-1<y\). The value of \(x\) is chosen using the given \(y\), which is permitted by the stated order.
The second statement is false. If such a real number \(x\) existed, the condition would require \(x<y\) for every real \(y\). In particular, it would require \(x<x\), by taking \(y=x\). But no real number is strictly less than itself. Therefore no such \(x\) exists. The two sentences differ only in quantifier order, but one is true and the other is false.
Precision About Boundaries and Alternatives
Small wording choices can change the set of cases covered by a sentence. “Positive” means strictly greater than zero; “nonnegative” permits zero. Likewise, \(x< a\) does not include \(a\), whereas \(x\leq a\) does. If a proof establishes a weak inequality, writing a strict inequality in the conclusion makes the claim stronger than the argument supports.
When a sentence includes alternatives, make clear whether either alternative is allowed, whether both are allowed, or whether exactly one is intended. In ordinary speech, “or” sometimes suggests one choice to the exclusion of the other. In standard mathematical logic, \(P\lor Q\) is true when at least one of \(P\) and \(Q\) is true, including when both are true. If the exclusive meaning is required, write it explicitly.
| Less precise wording | Question to resolve | More explicit form |
|---|---|---|
| “A number is positive.” | Which number and which domain? | “The real number \(x\) satisfies \(x>0\).” |
| “A number has a solution.” | Is the claim about one number or every number? | “For every \(a\in D\), there exists \(x\in E\) satisfying the equation.” |
| “\(P\) and then \(Q\).” | Is \(P\) a condition that implies \(Q\), or do both hold? | “If \(P\), then \(Q\),” or “\(P\land Q\),” as intended. |
| “At least zero.” | Is equality allowed? | Use \(x\geq0\) for nonnegative and \(x>0\) for positive. |
An important final check is whether every symbol has been introduced and every pronoun has a clear referent. A statement such as “If \(a<b\), then it is less than \(c\)” is incomplete unless “it” clearly names an object. Write the object again: “If \(a<b\), then \(a<c\).” Repeating a variable is usually preferable to forcing a reader to infer its identity.
A Short Revision Process
A sentence can be revised systematically. First, identify what the sentence is meant to assert, without changing its strength. Then expose the domain and quantifiers. Next, mark the logical connection between the conditions. Finally, test the resulting sentence on a simple example or boundary case. This process often reveals that a vague phrase was hiding an unstated assumption or a missing alternative.
Introduce each variable and state its domain, such as \(x\in\mathbb R\) or \(n\in\mathbb Z\).
Decide whether the claim concerns every object, at least one object, or a sequence of dependent choices.
Decide whether the sentence asserts an implication, a conjunction, an alternative, or an equivalence.
Verify which variable each condition concerns and whether zero, equality, or both alternatives are included.
Ask whether the written statement is exactly the claim intended, and whether a simple test case exposes an ambiguity.
These steps are not a demand to translate every sentence into symbols. Mathematical prose remains valuable, especially when explaining why a statement is true. The aim is to make the prose carry the same definite meaning as a correctly written symbolic form. Once the claim is precise, a proof can be planned for that claim rather than for an informal approximation of it.
Check Your Understanding
For each question, focus on what the wording says and how you would make its meaning explicit.
- Rewrite “A number has a larger number” as a statement about every real number, and give a choice of the larger number for an arbitrary input.
- In the sentence “For every \(y\), there is an \(x\) with \(x<y\),” may the choice of \(x\) depend on \(y\)? How does that differ from putting “there is an \(x\)” first?
- Write “Every integer divisible by \(10\) is divisible by \(5\)” with an explicit domain and conditional structure.
- Explain the difference between “nonnegative” and “positive,” and give a real number that is nonnegative but not positive.
- A proof uses the phrase “it follows.” What should you check about the sentence immediately before it and the conclusion that follows?
- What information should be made explicit before beginning a proof of a sentence involving two variables and two quantifiers?