Statements and Definitions
A proof is a sequence of statements, each of which is an assumption, a definition, a previously proved fact, or a logical consequence of earlier lines, ending at the claim. Nothing else is allowed — not examples, not diagrams that "look right", not "it is obvious". The reader must be able to check every step without trusting you.
The strange thing about proofs, for someone meeting them for the first time, is how little they resemble the way we are convinced of things in daily life. Checking twenty cases convinces; it does not prove, because the twenty-first might fail. What a proof provides is a reason that covers every case at once. The reason usually comes from a definition: to prove something about even numbers, you will write , because that is what "even" means, and everything follows from that sentence.
Even: . Odd: . : . Rational: with integers , . Prime: with no divisors other than and . Every proof about these objects begins by unpacking the definition.
Claim: the sum of two odd numbers is even. Checking: , — convincing, but not a proof. Proof: let the numbers be and , where and are integers (the definition of odd, with a different letter for each number). Their sum is . Since is an integer, the sum is times an integer, which is the definition of even.
Claim: the square of every odd integer leaves remainder when divided by . Proof. Let with an integer. Then . Of the two consecutive integers and one is even, so for some integer . Therefore .
Each line has a reason. The first is the definition of odd. The second is algebra, and the factoring is chosen on purpose: it exposes a product of consecutive integers, which we know something about. The third line is itself a small proof by cases (if is even, is the even one; if is odd, is). Checking , and is how you would guess the claim; the factoring is what proves it.
Using the same letter for two different unknowns. "Let the odd numbers be and " proves only that an odd number added to ITSELF is even. Two numbers that are not given as equal need two letters.
Prove that the product of two odd numbers is odd.
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Let the numbers be and with integers. Their product is . Since is an integer, the product is times an integer plus , which is odd.
Statements and Implications
A statement is a sentence that is either true or false. The implication "if then " (written ) is false in exactly one situation: true and false. When is false the implication makes no promise, and it counts as true.
Think of an implication as a promise: "if you finish the problem set, you may leave early". The promise is broken only if you finish and are kept in; if you never finish, nothing was promised. Mathematics needs this convention for a sharper reason. The theorem "for every integer , if then is even" should be true — yet for the hypothesis is false. If a false hypothesis made the implication false, every theorem with a hypothesis would be "refuted" by the numbers it is not about. So a counterexample to must make true and false; nothing else counts.
From come three relatives: the converse , the inverse , and the contrapositive . The contrapositive always has the same truth value as the original. The converse and inverse need not — they are each other's contrapositives, so they stand or fall together.
"If is divisible by , then is divisible by ." True: . Converse: "if then " — false, because has and . Contrapositive: "if then " — true, as it must be, since it says the same thing as the original.
"If then " is false: makes the hypothesis true and the conclusion false, which is exactly the shape a counterexample must have. There are two honest repairs. Strengthen the hypothesis: "if and then ". Or weaken the conclusion: "if then or ". Both are true, and the second is even an "if and only if".
"If a number is prime, then it is odd" really means "for every integer , if is prime then is odd". A counterexample needs one that is prime and not odd: . The numbers (odd, not prime) and (even, not prime) are not counterexamples — their hypothesis is false, so the implication makes no claim about them. Finding the right kind of counterexample is half the skill of reading a statement.
Using a converse as if it had been proved. Inside a longer argument this is usually silent: " and , so " treats "if then and " as reversible. It is not: is divisible by and by but not by . The step is valid only when the two divisors share no common factor.
Statement: "If a quadrilateral is a square, then its diagonals are equal." Write the converse, the inverse and the contrapositive, and say which of the four are true.
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Original: true. Converse: "if a quadrilateral's diagonals are equal, it is a square" — false; a rectangle has equal diagonals. Inverse: "if a quadrilateral is not a square, its diagonals are not equal" — false, by the same rectangle (the inverse is the converse's contrapositive, so they fail together). Contrapositive: "if the diagonals are not equal, it is not a square" — true, because it is equivalent to the original.
Reading a Proof
Read a proof three times. First, find the claim and the assumptions. Second, at each line, ask "why is this true?" and do not move on until you can say. Third, ask "where was each assumption used?" — if some assumption was never used, either the theorem is stronger than stated or the proof is wrong.
Mathematicians read slowly. A page of proof may take an hour, most of it spent with a pencil checking a line the author called "clearly". That checking is the reading; a proof skimmed is a proof not read. The third pass — where were the hypotheses used? — is the most instructive one, because it shows what the proof is really about and what would break without each assumption.
Theorem: if is even then is even. Proof by contrapositive: suppose is odd, . Then is odd. So if is even, cannot be odd. Where is " even" used? Only at the end, as the thing being contradicted. The real work is the algebra showing odd squares are odd.
Claim: the product of four consecutive integers, plus , is a perfect square. The printed proof says: "." A reader does not accept this; a reader checks it. Pair the outer and inner factors: and . With the left side is , which is the claim. A number test catches a misprint before the algebra does: gives and .
Reading for plausibility instead of for reasons. A line like "since , " looks routine and is false in general (, ). It is correct only if the proof has already established — so the reader's job is to find where that was established, or to flag the gap.
Find the error: "Claim: if , then or . Proof: since divides , shares a factor with or with , so it divides one of them."
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The claim is false: , but and . The broken step is "shares a factor, so it divides": shares the factor with , which does not make a divisor of . The statement becomes true when is a prime (Euclid's lemma), and its proof needs more than this sentence.
Putting It Together
Claim: for every integer , is even.
Draft A. ", , , : all even, so is always even." Flaw: this checks four cases and says nothing about . No reason is given that covers every .
Draft B. "Let be an integer. Then , a product of two consecutive integers. Of any two consecutive integers one is even: if is even it is , and if is odd then is even. A product with an even factor is even. "
Draft B is a proof because every sentence is a definition, a line of algebra, or a fact with its reason attached, and because is arbitrary — nothing about it was assumed except that it is an integer.
Before writing: state the claim with its quantifiers and unpack every definition. While writing: each line is an assumption, a definition, a proved fact, or follows from earlier lines — and says which. After writing: the last line is the claim, word for word.
Prove that is even for every integer .
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. If is even, the product has an even factor. If is odd, and is even. Either way the product is even. (Alternatively: , a product of consecutive integers plus an even number.)