From Reading Statements to Establishing Them
In Understanding Quantifier Order, we distinguished a witness chosen separately for each input from one witness that works for all inputs. The arguments there did more than identify truth values: they explained why the claims followed from the meanings of the quantifiers and the properties of the objects involved.
We now examine that kind of explanation itself. Understanding what a statement says is the first task. A proof supplies the logical justification for accepting it.
A proof need not express every step in logical symbols. In ordinary mathematical writing, sentences explain the roles of variables and the reasons for deductions, while equations record calculations. What matters is that the reasoning can be checked, not that the page contains many symbols.
What May a Proof Use?
Mathematics does not begin each argument from nothing. It builds on an established foundation. The following terms distinguish the roles of statements within that foundation.
| Term | Role in mathematical reasoning |
|---|---|
| Definition | Specifies the meaning of a term or notation. |
| Axiom | Is accepted as a starting assumption within the mathematical framework. |
| Theorem | Is a statement established by proof. |
| Conjecture | Is a proposed statement believed to be true but not yet proved. |
The words proposition, lemma, and corollary also commonly label proved results. A lemma usually supports another result; a corollary follows readily from an established result. These labels describe how a result is used, not a weaker standard of justification.
In the elementary examples below, we use the usual arithmetic and order properties of real numbers. For instance, a product of two positive real numbers is positive. Such a property may be invoked without reconstructing all of real arithmetic each time it is needed.
When using a previously proved theorem, however, its hypotheses must be satisfied. A theorem about positive numbers cannot be applied to an arbitrary real number without first checking positivity.
Hypotheses, Conclusions, and Scope
Consider a statement of the form
Here \(D\) is the domain, \(P(x)\) is the hypothesis of the implication, and \(Q(x)\) is its conclusion. The claim is that every input in the domain satisfying \(P(x)\) also satisfies \(Q(x)\).
For an input satisfying \(P(x)\), a proof must justify \(Q(x)\). For an input not satisfying \(P(x)\), the implication is true by the meaning of implication established earlier in this course. Thus assuming the hypothesis while proving a conditional claim does not assume the conclusion.
The phrase “let \(x\in D\) be arbitrary” also carries a precise obligation. The argument must not give \(x\) a special property beyond membership in \(D\) and any stated hypotheses. It is this absence of extra restrictions that allows the conclusion to apply to every permitted input.
For example, choosing \(x=2\) does not represent all real numbers greater than \(1\). An arbitrary \(x>1\) might be \(3/2\), \(10\), or any other real number satisfying the hypothesis. A proof must apply without depending on which such value is used.
A Complete Proof and Its Justifications
Proof. Let \(x\in\mathbb R\) be arbitrary and suppose \(x>1\). Then \(x-1>0\) and \(x+1>2>0\). Since a product of positive real numbers is positive,
Adding \(1\) gives \(x^2>1\). This proves the implication for every real \(x>1\); for other real inputs the hypothesis is false, so the implication is true. Therefore the stated universal claim holds.
The proof is short, but its steps have distinct roles.
| Part of the proof | Why it is justified |
|---|---|
| Fix arbitrary real \(x\) with \(x>1\). | These are exactly the domain and hypothesis in the claim. |
| \(x-1>0\) and \(x+1>0\). | Addition preserves inequalities; \(x+1>2>0\). |
| \(x^2-1=(x-1)(x+1)\). | The identity follows from the distributive law. |
| \(x^2-1>0\). | Both factors have been shown to be positive. |
| \(x^2>1\). | Add \(1\) to the preceding inequality. |
No numerical sampling is needed. The calculation applies to every input satisfying the hypothesis because it uses only properties shared by all those inputs.
The proof establishes exactly the stated implication. It does not establish its converse, “if \(x^2>1\), then \(x>1\).” That converse is false: \(x=-2\) satisfies \(x^2=4>1\) but not \(x>1\).
Examples, Witnesses, and Counterexamples
An example can suggest a claim, but its logical force depends on the claim’s quantifiers. The same single calculation may be enough for an existence statement and insufficient for a universal statement.
Worked Example: What One Calculation Establishes
The calculation \(2^2=4>2\) establishes
Indeed, \(x=2\) is a real witness and satisfies the required inequality. It does not establish
The universal statement is false: \(x=0\) is an allowed input, and \(0^2=0\), so the strict inequality fails.
A counterexample to a universal claim is an allowed input at which its asserted condition is false. By the quantifier-negation rules established earlier,
This is why one verified counterexample disproves a universal claim. For a conditional universal claim, the counterexample must satisfy the hypothesis and fail the conclusion. An input that violates the hypothesis does not refute the implication.
Examples remain useful for exploring a conjecture and detecting errors. Their limitation is not that they lack value, but that their conclusions must not exceed what they actually verify.
A Proof Must Match the Quantifiers
The previous tutorial showed that even correct arithmetic can support the wrong statement if the variables are chosen in the wrong order. A proof must satisfy both the final condition and the requirements imposed by the quantifiers.
Worked Example: Existence for Each Input
We claim
Proof. Let \(x\in\mathbb R\) be arbitrary. Set \(y=(3-x)/2\). This is a real number because real arithmetic is closed under subtraction and division by the nonzero real number \(2\). Moreover,
Thus an allowed witness exists for every real \(x\), proving the claim.
The witness is allowed to depend on \(x\), since \(x\) is fixed first. This argument does not supply one \(y\) that works for every real \(x\). In fact, such a common witness cannot exist: \(x=0\) would require \(y=3/2\), while \(x=1\) would require \(y=1\).
The proof has three essential features: it introduces the arbitrary input, supplies a witness in the required domain, and verifies the stated condition. Merely writing down a plausible formula for \(y\) would leave the verification unfinished.
Recognizing Gaps in an Argument
A true conclusion does not make every argument for it valid. Proof concerns the connection between the starting information and the conclusion, not just the conclusion’s truth value.
Worked Example: An Unjustified Division
Consider the attempted argument: “If a real number \(x\) satisfies \(x^2=x\), divide by \(x\) to obtain \(x=1\).”
Division by \(x\) requires \(x\neq0\), which has not been established. The omitted value \(x=0\) actually satisfies \(x^2=x\), so the asserted conclusion is false.
A valid conclusion is that \(x=0\) or \(x=1\). To establish it, suppose \(x^2=x\). If \(x=0\), the conclusion already holds. If \(x\neq0\), division by \(x\) is permitted and gives \(x=1\). These alternatives cover all real \(x\). Both \(0\) and \(1\) satisfy the equation by substitution, so they are exactly its real solutions.
Another common gap is circular reasoning: using the desired conclusion, directly or through an equivalent unproved assertion, as its own justification. For example, “\(x^2>1\) because \(x^2-1>0\)” does not prove anything unless the latter inequality has been established independently. In the theorem above, the positive factorization provided that missing justification.
Exploration and presentation also differ. While searching for an argument, one may manipulate the desired conclusion to discover a useful identity. The finished proof must still explain why the conclusion follows from available information; the discovery process alone does not supply that logical connection.
A Checklist for Reading a Proof
Read the domains, quantifiers, hypotheses, and conclusion before following the calculations.
Distinguish arbitrary inputs, assumed hypotheses, chosen witnesses, and facts already established.
Ask which definition, arithmetic property, logical rule, or earlier theorem justifies it, and whether its hypotheses hold.
Check that the argument covers every required input, verifies every witness, and proves the intended direction.
A proof is complete when these obligations have been met, not when the calculation looks persuasive. This standard allows later results to build reliably on earlier ones: once a statement is proved, it becomes available for further reasoning whenever its hypotheses are satisfied.
Check Your Understanding
Give a justification for each answer, and distinguish what is assumed from what must be established.
- In “For every real \(x\), if \(x>3\), then \(x+2>5\),” identify the domain, hypothesis, and conclusion. Explain why taking only \(x=4\) does not prove the statement.
- The calculation \(3^2>3\) verifies a particular inequality. Write an existential statement it proves. Give a counterexample to the corresponding universal statement over \(\mathbb R\).
- Explain why \(x=0\) does not refute the theorem “if \(x>1\), then \(x^2>1\).” What two conditions must an input satisfy to refute an implication?
- Prove \(\forall x\in\mathbb R,\ \exists y\in\mathbb R,\ x+3y=6\). Identify where you check the witness’s domain and where you verify the equation.
- An argument says, “If \(ab=a\), divide by \(a\), so \(b=1\).” Identify the missing hypothesis, give a counterexample to the claim as written, and state a corrected claim.
- A proposed proof of \(x^2>1\) from \(x>1\) says only, “The conclusion holds because \(x^2-1>0\).” What justification is missing? Supply it without assuming the conclusion.