A Definition Is a Precise Instruction
A mathematical definition fixes the meaning of a term. It tells us exactly what conditions an object must satisfy for that term to apply. The definition itself is not a proof that any particular object satisfies those conditions. It gives us a precise test to use when we examine an object or construct an argument.
Earlier in this course, definitions such as evenness were used as starting points in proofs. When a proof says “by the definition,” the next step is to expand the defined term into its stated condition. That is why reading a definition carefully is an essential part of reading a proof: a small change in the domain, a quantifier, or a comparison can change the claim.
Consider the word “upper bound.” In everyday speech, “upper” might suggest a value near the top, or perhaps a largest value. The mathematical definition is more exact and does not require an upper bound to be an element of the set. Let \(A\) be a subset of \(\mathbb R\).
The first sentence defines when a particular real number \(M\) is an upper bound for a particular set \(A\). The second defines a property of the set: being bounded above means that some such \(M\) exists. These are related but not interchangeable statements. One names a candidate bound; the other asserts that a candidate can be found.
Unpack the Structure Before Applying It
A reliable first pass through a definition is to identify the objects it talks about, the domain from which each object comes, and the logical structure connecting the conditions. In the upper-bound definition, \(A\) is a set of real numbers, \(M\) is a real number, and the condition must hold for every element \(x\) of \(A\). Written with quantifiers, the definition becomes:
$$ M\text{ is an upper bound for }A \quad\Longleftrightarrow\quad M\in\mathbb R\text{ and }\forall x\in A,\ x\leq M. $$If the domain information \(M\in\mathbb R\) is already understood, the main condition is \(\forall x\in A,\ x\leq M\). To verify it, take an arbitrary element of \(A\) and show that it is no larger than \(M\). To show that \(M\) is not an upper bound, it is enough to find an element \(x\in A\) with \(x>M\).
For “bounded above,” the quantifier structure has an additional existence step: first choose a real number \(M\), then require it to work for every \(x\in A\).
$$ A\text{ is bounded above} \quad\Longleftrightarrow\quad \exists M\in\mathbb R\ \forall x\in A,\ x\leq M. $$The order of these quantifiers matters. One fixed \(M\) must work for all elements of \(A\). It is not enough that each element has some real number above it, since the upper-bound condition asks for a shared bound for the entire set.
Record what the definition is about: a number, a set, a function, or another mathematical object.
Notice whether variables must be real numbers, integers, members of a set, or inputs to a specified function.
Translate “every,” “some,” and “if and only if” into the corresponding quantifiers and implications.
Write down exactly what must be shown for the term to apply to the object in question.
For verification, establish all required conditions; for failure of a universal condition, find a permitted input where it fails.
Worked Example: Checking an Upper Bound
Let \(A=\{2,5,8\}\). We will check whether \(M=8\) is an upper bound for \(A\). The definition requires that each element of \(A\) be at most \(8\). The three checks are $$ 2\leq8,\qquad 5\leq8,\qquad 8\leq8. $$ All hold, so \(8\) is an upper bound for \(A\). Consequently, \(A\) is bounded above: the number \(8\) witnesses the required existence.
The same set also has \(10\) as an upper bound, because \(2\leq10\), \(5\leq10\), and \(8\leq10\). The definition asks for a number at least as large as every element; it does not require the bound to be the smallest possible one. In contrast, \(M=7\) is not an upper bound: \(8\in A\), but \(8\leq7\) is false.
“If and Only If” Gives Two Directions
Many definitions use the phrase “if and only if.” This says that the term applies exactly when the stated condition holds. It contains two implications: the defined term implies the condition, and the condition implies that the term applies. In a proof, the direction you need depends on what is given and what you are trying to establish.
For example, suppose a function \(f:D\to E\) is called injective according to the following definition.
The statement is not that every pair \(x,y\in D\) must be equal. The condition is conditional: whenever two outputs are equal, their inputs must be equal. To prove injectivity, begin with arbitrary \(x,y\in D\), assume \(f(x)=f(y)\), and deduce \(x=y\). To show that \(f\) is not injective, find two distinct inputs with equal outputs.
Worked Example: Verifying Injectivity from the Definition
Define \(f:\mathbb R\to\mathbb R\) by \(f(x)=4x+3\). We check the defining condition directly. Let \(x,y\in\mathbb R\) and suppose \(f(x)=f(y)\). Then $$ 4x+3=4y+3. $$ Subtracting \(3\) from both sides gives \(4x=4y\). Dividing by the nonzero number \(4\) gives \(x=y\). Since this argument applies to arbitrary real numbers \(x\) and \(y\), the definition shows that \(f\) is injective.
The key is the direction of reasoning: equality of outputs is the assumption, and equality of inputs is the conclusion. Merely observing that \(f(x)\) has a formula does not establish the defining condition.
Definitions and theorems also play different roles in an argument. A definition tells us what a term means; a theorem is a mathematical statement established by a valid proof, as explained in What Is Mathematics? A definition can be used in a proof, but introducing a definition does not establish that examples exist or that a further property follows.
| What you see | How to read it | What a proof might require |
|---|---|---|
| “For every \(x\in D\)” | The condition applies to each permitted input. | Take an arbitrary \(x\in D\) and establish the condition. |
| “There exists \(M\in\mathbb R\)” | At least one real number must work. | Give a candidate \(M\) and verify its required property. |
| “If \(P\), then \(Q\)” | \(P\) is the hypothesis; \(Q\) is the conclusion. | Assume \(P\) and deduce \(Q\). |
| “If and only if” | Both directions hold. | Establish each implication, unless one is already available. |
How to Negate a Definition-Based Claim
Sometimes a definition is easiest to understand by asking what failure would look like. The rules for negating universal and existential statements were established earlier in Negating Universal Statements and Negating Existential Statements. Applying those rules to the upper-bound definition gives a precise test for a set that is not bounded above.
The assertion that \(A\) is bounded above is \(\exists M\in\mathbb R\ \forall x\in A,\ x\leq M\). Its negation is $$ \forall M\in\mathbb R\ \exists x\in A,\ x>M. $$ Thus a set is not bounded above exactly when every proposed real bound can be exceeded by some element of the set. Notice the order: for each proposed \(M\), the element \(x\) may depend on \(M\). The same element is not required to exceed every real number.
Worked Example: Showing a Set Is Not Bounded Above
Let \(A=\{x\in\mathbb R:x\geq0\}\). To show that \(A\) is not bounded above, let \(M\in\mathbb R\) be arbitrary. Choose $$ x=|M|+1. $$ Since \(|M|\geq0\), we have \(x\geq1>0\), so \(x\in A\). Also \(M\leq|M|\), and therefore $$ x=|M|+1>|M|\geq M. $$ For every proposed real number \(M\), we have produced an element \(x\in A\) with \(x>M\). The negation of the bounded-above condition is satisfied, so \(A\) is not bounded above.
This argument follows the quantifiers in their stated order: start with an arbitrary proposed bound, then construct an element that exceeds it. Choosing one large number without accounting for an arbitrary \(M\) would not establish the claim.
Two Consequences of Reading the Definition
Once the definition of upper bound is explicit, some useful facts follow by applying its condition directly. These are results, not additional parts of the definition. The first describes how upper bounds behave when the candidate is increased.
Proof. Let \(x\in A\) be arbitrary. Since \(M\) is an upper bound for \(A\), the definition gives \(x\leq M\). By hypothesis, \(M\leq N\). Transitivity of the order on \(\mathbb R\) gives \(x\leq N\). This holds for every \(x\in A\), so \(N\) is an upper bound for \(A\) by definition. If \(A\) is empty, there are no elements \(x\in A\) to check, and the universal condition holds as well. This proves the proposition in all cases.
A second consequence concerns inclusion of sets. If every element of one set is also an element of another, then any upper bound for the larger set works for the smaller set.
Proof. Since \(B\) is bounded above, there exists \(M\in\mathbb R\) such that \(y\leq M\) for every \(y\in B\). Let \(x\in A\). The inclusion \(A\subseteq B\) gives \(x\in B\), so \(x\leq M\). This holds for every \(x\in A\), and the same real number \(M\) is therefore an upper bound for \(A\). Hence \(A\) is bounded above. If \(A\) is empty, the condition for \(M\) to bound \(A\) holds vacuously; the argument remains valid.
Both proofs illustrate a general method: choose an arbitrary object covered by the definition, apply the defining condition, and then use the hypotheses to reach the required conclusion. The definition is not merely vocabulary to memorize; it provides the exact steps a proof needs.
Check Your Understanding
Use the definitions and methods in this tutorial to answer each question.
- In the definition of an upper bound for \(A\subseteq\mathbb R\), what must be true of a candidate \(M\) for every \(x\in A\)?
- For \(A=\{1,4,6\}\), is \(5\) an upper bound? Identify an element that verifies your answer.
- State the quantifier form of “\(A\) is bounded above.” Which number must work for all elements of \(A\)?
- To prove that \(f:D\to E\) is injective, what should you assume about \(f(x)\) and \(f(y)\), and what must you deduce?
- Write the logical condition that shows \(A\subseteq\mathbb R\) is not bounded above.
- If \(A\subseteq B\) and \(M\) is an upper bound for \(B\), explain why the same \(M\) is an upper bound for \(A\).