Writing Definitions Precisely

A Definition Fixes Meaning

The previous tutorial, Writing Precise Mathematical Sentences, focused on making claims clear: identify the objects, specify their domains, and make the logical connections explicit. A definition uses those same tools for a particular purpose. It introduces a term by saying exactly which objects the term describes.

For example, we can introduce a name for a real number satisfying a chosen condition: “A real number $x$ is a quarter-integer if there exists an integer $n$ such that $x=n/4$.” This definition tells us the domain of $x$, the condition that qualifies it, and the meaning of the new term. Once the meaning is fixed, a statement about quarter-integers can be proved or disproved using that condition.

A definition is not a theorem. A definition assigns a meaning; it is not something to prove true. A theorem is a claim established by proof from definitions, axioms, and earlier results, as discussed in the first tutorial. We may, however, prove consequences of a definition. For instance, after defining a term, we can prove that particular objects do or do not satisfy it.

A definition should let a reader decide membership. Given an object in the stated domain, the defining condition should determine whether the term applies. If the reader must guess what “small,” “simple,” or “appropriate” means, the definition has not yet supplied a mathematical criterion.

State the Domain and the Full Criterion

A definition needs a domain: the collection of objects to which the new term applies. “A number is a quarter-integer” leaves open whether “number” means an integer, a rational number, or a real number. The domain matters even when the condition looks familiar. A definition of a property of real numbers should say “a real number $x$,” while a definition of a property of sets should identify the sets being discussed.

The defining condition must also be complete. If the term depends on the existence of another object, state both the domain of that object and the required relationship. In the quarter-integer definition, $n$ must be an integer; it is not enough to say that $x$ is “one fourth of a number.” The intended condition is $$ \exists n\in\mathbb Z,\quad x=\frac{n}{4}. $$ This says that at least one integer $n$ represents $x$ in the required form. It does not require that $n$ be unique.

When the term is defined by a condition $P(x)$, the phrase “$x$ is called [term] if $P(x)$” is often convenient. In a definition, “if” introduces the criterion for using the term; it is not a claim that needs to be proved. The symbols $P(x)\Longleftrightarrow$ “[term applies to $x$]” can express the same intended assignment of meaning, provided the domain is clear.

Worked Example: Testing a Quarter-Integer Definition

Use the definition: a real number $x$ is a quarter-integer if there exists $n\in\mathbb Z$ such that $x=n/4$. Determine whether $7/4$ and $2/3$ are quarter-integers.

For $x=7/4$, choose $n=7$. Since $7\in\mathbb Z$ and $7/4=n/4$, the defining condition holds, so $7/4$ is a quarter-integer.

Suppose, for the other number, that $2/3$ were a quarter-integer. Then there would be an integer $n$ such that $$ \frac{2}{3}=\frac{n}{4}. $$ Multiplying both sides by $4$ gives $n=8/3$. But $8/3$ is not an integer, so no such $n\in\mathbb Z$ exists. Therefore $2/3$ is not a quarter-integer. The second conclusion requires ruling out every possible integer witness, not merely failing to find one by inspection.

Definitions Involving Sets and Relations

Sometimes a definition names a collection of objects that satisfy a condition. The set-builder notation $$ \{x\in D:P(x)\} $$ means the set of all $x$ in the domain $D$ for which the condition $P(x)$ is true. Both parts matter: $x\in D$ limits the allowed objects, and $P(x)$ says which of those objects are included. Omitting the domain can change the set or leave its meaning unclear.

Other definitions describe a relationship between multiple objects. For example, “$x$ lies in the open interval from $a$ to $b$” is a condition on $a$, $b$, and $x$, and it includes the requirement $a<x<b$. For this to describe the usual open interval with its endpoints in the intended order, specify $a,b\in\mathbb R$ with $a<b$. Then define $$ (a,b)=\{x\in\mathbb R:a<x<b\}. $$ The displayed condition says that both strict inequalities must hold. Neither endpoint is included.

A definition can also use quantifiers to describe a property of a set. For instance, a useful definition of boundedness for a subset $A$ of $\mathbb R$ is that there exists a real number $M>0$ such that every $x\in A$ satisfies $|x|\leq M$. The order of quantifiers matters: one $M$ must work for all elements of $A$, rather than a different bound being chosen separately for each $x$.

Worked Example: Reading an Open-Interval Definition

Let $a=2$ and $b=5$. By definition, $$ (2,5)=\{x\in\mathbb R:2<x<5\}. $$ The real number $3$ belongs to $(2,5)$, because $2<3$ and $3<5$. The number $2$ does not belong to $(2,5)$, because the required strict inequality $2<2$ is false. The number $6$ also does not belong, because $6<5$ is false.

The condition consists of two inequalities joined together: an element must be greater than the left endpoint and less than the right endpoint. Replacing either strict inequality by a weak one would define a different set. For example, $\{x\in\mathbb R:2\leq x<5\}$ includes $2$, so it is not the open interval $(2,5)$.

Use Definitions That Do Not Depend on Themselves

A definition should explain its term using language whose meaning is already understood. A circular definition does not make that meaning available. For example, “A real number is substantial if it is a number of substantial size” repeats the undefined idea rather than providing a criterion. A more useful definition would specify a measurable condition, such as $x>10$, if that is the intended meaning.

An informal word is not automatically unsuitable. It can be introduced if its meaning is made exact. But if the definition relies on words such as “near,” “large,” or “regular,” the relevant threshold or condition must be stated. “Near $c$” could mean $|x-c|<1$, $|x-c|<0.01$, or something else. Without a specified criterion, different readers may reasonably classify the same $x$ differently.

Also distinguish a condition from a description that merely gives examples. Saying “a special number is $1$, $3$, or $5$” defines a finite set if those are exactly the intended objects. Saying “a special number is like $1$, $3$, or $5$” gives no precise test for other values. A definition should cover all cases, including cases not listed as examples.

Worked Example: Making a Set Definition Complete

Consider the phrase “a real number is close to $c$.” As written, it does not determine a definite set: the permitted distance has not been specified. Suppose the intended meaning is “within one unit of $c$, including the endpoints.” A precise definition is: for $c\in\mathbb R$, $x$ is within one unit of $c$ if $|x-c|\leq1$.

For $c=4$ and $x=5$, we have $|5-4|=1$, so $x$ satisfies the definition because equality is allowed. For $x=6$, $|6-4|=2>1$, so it does not. If the intended phrase had meant “strictly less than one unit away,” the definition would instead use $|x-c|<1$, and $x=5$ would fail that version. Stating whether equality is included removes the ambiguity.

Equivalent Criteria Give the Same Class

Different definitions can describe the same collection of objects. One may be more convenient for a proof, while another may make a different feature easier to see. To justify that two criteria give the same meaning on a fixed domain, show that they are equivalent for every object in that domain. This is a useful way to check a proposed alternative definition without relying on the wording alone.

Here is the precise statement. Let $D$ be a domain, and let $P(x)$ and $Q(x)$ be conditions defined for every $x\in D$. Define $$ A=\{x\in D:P(x)\},\qquad B=\{x\in D:Q(x)\}. $$ If $P(x)\Longleftrightarrow Q(x)$ for every $x\in D$, then $A=B$.

Proof. We prove that the two sets have the same elements. Let $x\in A$. By the definition of $A$, $x\in D$ and $P(x)$ holds. The assumed equivalence on $D$ then gives $Q(x)$. Since $x\in D$ and $Q(x)$, the definition of $B$ gives $x\in B$. Thus $A\subseteq B$.

For the reverse inclusion, let $x\in B$. Then $x\in D$ and $Q(x)$ holds. The equivalence gives $P(x)$, so the definition of $A$ gives $x\in A$. Thus $B\subseteq A$, and consequently $A=B$. The domain condition is essential: the equivalence was assumed for inputs in $D$, and the definitions use that same domain.

This result offers a practical standard for checking a rewritten definition: the old and new criteria must agree on every allowed object. Agreement on a few examples is useful evidence for finding mistakes, but it does not prove that the criteria always agree.

Restricting a Definition's Domain

Changing the domain can change the collection described by a condition, even if the condition itself is unchanged. The relationship is exact: restricting the domain from a larger set to a smaller set keeps precisely the members of the original class that lie in the smaller domain.

Proposition. Let $D\subseteq E$, and let $P(x)$ be a condition defined for every $x\in E$. Set $$ A_D=\{x\in D:P(x)\},\qquad A_E=\{x\in E:P(x)\}. $$ Then $A_D=D\cap A_E$.

Proof. Let $x$ be arbitrary. If $x\in A_D$, then $x\in D$ and $P(x)$ holds. Because $D\subseteq E$, we also have $x\in E$. The condition $P(x)$ and membership $x\in E$ imply $x\in A_E$. Hence $x\in D\cap A_E$.

Conversely, suppose $x\in D\cap A_E$. Then $x\in D$, and $x\in A_E$ means $x\in E$ and $P(x)$ holds. In particular, $x\in D$ and $P(x)$, so $x\in A_D$. We have proved that $x\in A_D$ exactly when $x\in D\cap A_E$. Therefore $A_D=D\cap A_E$.

For example, the condition $x^2=1$ describes $\{-1,1\}$ when the domain is $\mathbb R$. If the domain is instead the nonnegative reals, the defined collection is $\{1\}$: the condition has not changed, but the allowable inputs have. This is why a definition's domain is part of its meaning, not an optional preface.

A Practical Definition-Writing Check

Before using a definition, read it as a decision procedure. Can you tell what kinds of objects may be tested? Can you decide what condition an object must meet? If the definition introduces another variable, do you know its domain and whether it is chosen once or separately for each input? Finally, test the definition on an ordinary example and an edge case. These checks help expose missing quantifiers, unclear boundaries, and unintended inclusions.

Name the object and its domain.
Specify whether the term applies to real numbers, integers, sets, or another stated collection.

Write the exact condition.
Replace vague descriptions with equations, inequalities, or other criteria that can be checked.

Make quantifiers explicit.
State whether a witness exists, whether a condition must hold for every object, and which choices may depend on earlier inputs.

Check boundaries and edge cases.
Determine whether equality is allowed and whether the domain includes any exceptional or empty cases.

Test the meaning.
Apply the definition to examples and ask whether two readers would classify each object in the same way.

Definitions are often concise, but concision should come after completeness. A symbol such as $\exists n\in\mathbb Z$ can efficiently express a choice, and set-builder notation can compactly name all objects satisfying a condition. Both are precise only when their domains and conditions are clear. The goal is not to make every definition elaborate; it is to make sure no essential part is left to interpretation.

A precise definition settles both who is eligible and what qualifies. Keep the domain fixed, state the complete condition, and check the edge cases. To show that two formulations define the same term, prove that their conditions are equivalent throughout that domain.

Check Your Understanding

For each question, focus on the domain, the defining condition, and what must be checked to use the term correctly.

A proposed definition says, “A real number $x$ is a fifth-integer if $x=n/5$ for some integer $n$.” Identify the domain of $x$ and the role of $n$. Is $3/5$ a fifth-integer?
Why is “$x$ is close to $c$” not a precise definition by itself? Give one precise condition that could express one possible meaning.
For $D\subseteq E$ and a condition $P$ on $E$, explain what the equality $\{x\in D:P(x)\}=D\cap\{x\in E:P(x)\}$ says about changing the domain.
A definition uses the condition $1\leq x<4$. Is the endpoint $1$ included? Is $4$? Explain using the inequalities.
What must be proved to show that two conditions define the same collection on a domain $D$?
Why does listing several examples of a property not, by itself, define how every other object is classified?

What You'll Learn