the sentences mathematics is allowed to argue about.
Mathematics does not argue about everything. It argues only about sentences that are definitely true or definitely false — and it calls those sentences statements. "Seven is prime" is a statement; you can firmly stamp it true. "Six is prime" is also a statement — it just gets stamped false. But "Is seven prime?", "Draw a circle", "π is beautiful", and "x is bigger than three" (with x left floating) are not statements: there is nothing there to call true or false yet. In this lesson we learn to tell statements from non-statements, to judge a statement by hunting for a single counterexample, and to read the most important shape in all of mathematics — "if p then q" — knowing exactly when it lies.
A statement is a sentence that has a definite truth value — it is either true or it is false, with no third option and no "it depends." That single requirement does a lot of work, because most sentences we say out loud are not statements.
Contrast those with "7 is prime" (definitely true) and "9 is prime" (definitely false). Both are genuine statements — being false does not disqualify a sentence; being undecidable in principle does.
A sentence is a statement exactly when you can, at least in principle, stamp it true or false — one or the other, definitely. Questions, commands, opinions, and open sentences fail that test.
Once a sentence is a statement, which value does it get? Many statements are claims about every member of a collection — "every prime is odd," "all squares are positive." A claim like that is true only if it survives every single case. It takes a perfect record to be true; it takes just one exception to be false.
That single exception has a name: a counterexample. Find one, and the universal claim collapses — no matter how many cases obeyed it. "Every prime is odd" sounds reasonable: 3, 5, 7, 11 … all odd. But the prime 2 is even. One counterexample, and the claim is false.
Claim: "every prime number is odd." We test the small primes 2, 3, 5, 7. Three of them — 3, 5, 7 — are odd and obey the claim. But 2 is prime and even, so it breaks the claim. Verdict: the statement is false, and the counterexample is x = 2. Notice we did not need to check every prime in existence — a single counterexample finished the job.
Checking lots of cases that work never proves a universal statement — there could always be one more case you have not reached. (We will make that precise with the quantifier ∀ in 35.3.) But a single counterexample does settle it: the claim is false, full stop.
The most useful statements in mathematics wear a special uniform: "if p, then q." We call p the hypothesis (the "if" part — what we assume) and q the conclusion (the "then" part — what we claim follows). We write it p ⇒ q, read "p implies q." Throughout this stage the hypothesis is blue and the conclusion is amber.
When is "if p then q" itself true? It is a promise: "whenever the hypothesis holds, the conclusion holds." A promise is broken only in one situation — the hypothesis came true but the conclusion did not. So:
| p (if) | q (then) | p ⇒ q |
|---|---|---|
| T | T | · |
| T | F | · |
| F | T | · |
| F | F | · |
Read the table: p ⇒ q is false in only one row — when p is true but q is false. In every other row it is true. The two bottom rows surprise people: when the hypothesis is false, the conditional is automatically true. We call this vacuously true — the promise was never tested, so it was never broken.
"If 0 = 1, then I am the king of France." Here the hypothesis 0 = 1 is false, so by the table the whole conditional is true — regardless of who is king. It feels like cheating, but the promise "whenever 0 = 1, then …" is never put to the test, so it is never broken. A conditional only earns a false when its hypothesis actually comes true and the conclusion still fails.
Most mathematical claims arrive in plain clothes — "multiples of 4 are even," "diagonals of a rectangle are equal." To reason about them, we put them in uniform: find the hidden hypothesis and conclusion, then write "if p then q."
The trick: a sentence of the form "every A is a B" (or "A's are B's") becomes "if something is an A, then it is a B."
| Everyday sentence | Standard form: if p then q |
|---|---|
| Multiples of 4 are even. | If n is a multiple of 4, then n is even. |
| Squares are non-negative. | If x is a real number, then x² ≥ 0. |
| A rectangle's diagonals are equal. | If a quadrilateral is a rectangle, then its diagonals are equal. |
Each rewrite makes the reasoning visible: the hypothesis is what you get to assume, and the conclusion is what you must establish. In 35.2 we will ask the natural next question — once a statement reads "if p then q," how does p relate to q? Is p enough on its own (sufficient), or merely required (necessary)?
"Every A is a B" rewrites as "if x is an A, then x is a B." Naming the hypothesis and conclusion turns a vague sentence into something you can prove — or disprove with one counterexample.
Four ideas open the whole stage.
| Idea | The rule |
|---|---|
| Statement | A sentence with a definite truth value — true or false. Questions, commands, opinions, open sentences are not statements. |
| True / false | A universal claim is true only with a perfect record; one counterexample makes it false. |
| p ⇒ q | p = hypothesis (blue), q = conclusion (amber). False only when p true and q false; vacuously true when p is false. |
| Rewriting | "Every A is a B" becomes "if x is an A then x is a B" — making the hypothesis and conclusion explicit. |
Which of these are statements? (a) "Is 12 even?" (b) "12 is even." (c) "Solve for x." (d) "x + 1 = 5" with x free.
Only (b) is a statement — it is definitely true. (a) is a question, (c) is a command, and (d) is an open sentence (true only for x = 4, undecided until x is fixed). None of those has a definite truth value.
Decide true or false, and if false give a counterexample: "Every even number is greater than 1."
False. The even number 0 is not greater than 1 (and neither are negative evens like −2). One counterexample, x = 0, settles it.
For "if n is a multiple of 6, then n is even," is the conditional true? What happens to the conditional when n = 9?
True. Whenever n is a multiple of 6 (6, 12, 18, …) the number is indeed even. At n = 9 the hypothesis "9 is a multiple of 6" is false, so the conditional is vacuously true there — it makes no promise about 9 at all.
Is "if it rains, then the ground is wet" made false by a day when it does not rain but the ground is wet anyway (someone watered the lawn)?
No — still true. The conditional is broken only when the hypothesis is true and the conclusion false (it rains, yet the ground stays dry). A dry-sky / wet-ground day has hypothesis false, so the promise was never tested. The conditional remains true.
Rewrite "Diagonals of a square bisect each other" in standard "if … then …" form, marking the hypothesis and conclusion.
"If a quadrilateral is a square, then its diagonals bisect each other." The hypothesis (blue) is what we assume; the conclusion (amber) is what we claim follows.
A student says: "I checked the claim 'every prime is odd' for 3, 5, 7, 11, 13 — all odd — so it's true." What is wrong?
Checking cases that work never proves a universal — you can always miss one. And here you do: the prime 2 is even, so the claim is false. A universal needs a perfect record; a single counterexample (x = 2) ends it.
Six questions to lock it in. Tap the answer you think is right.
This opening lesson of Stage 35 makes the truth value of a sentence the object of study: a statement is a sentence that is definitely true or false, and the conditional "if p then q" is false in exactly one case. It directly supports the Standards for Mathematical Practice MP3 (construct viable arguments and critique the reasoning of others) and MP6 (attend to precision), and it builds the if-then vocabulary that HSG-CO.C.9 relies on for writing and reading proofs. The recurring move — disproving a universal with a single counterexample, and recognizing vacuous truth — is exactly the precision students need before quantifiers (35.3) and proof (35.5).