"2+2=4" and "x+2=4" look almost identical, but only one of them is actually claiming something. The first one asserts a fact you can check and settle right now — it's true. The second one can't be checked until somebody says what x is; it's not making a claim at all yet, just holding a slot open. Discrete mathematics spends its first real habit on telling these two apart, because every later chapter assumes you can.
Think of a statement as a receipt. It's already been rung up. The total is fixed, so you can check it against reality. An open sentence is a blank receipt with a slot for the total. No amount is wrong yet, because none has been entered. Asking "is this receipt correct?" makes no sense until a number fills the blank. That's the trouble with x+2=4 on its own. Nobody can check it until x gets a value.
A statement is a declarative sentence that has exactly one of two truth values — true or false — with nothing left unresolved. "7 is prime" is a statement (true). "8 is prime" is a statement (false). "x is prime" is not a statement, because its truth depends on a value nobody has fixed; it's called an open sentence instead. A question ("is 7 prime?"), a command ("factor 7"), or an opinion ("7 is a nice number") also fail to be statements — not because they're false, but because "true or false" doesn't even apply to them. Statements can also be joined into bigger statements: "7 is prime and 8 is not" is still a single statement, with its own single truth value, built out of two smaller ones. Chapter 2 makes that joining precise; for now, the point is only that combining statements never produces an open sentence — the result is always still checkable.
Statement, open sentence, or neither
Sort each phrase by what it actually is: a statement with a fixed truth value, an open sentence waiting on a variable, or neither (a question or command).
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Notice card c5 above — a real, unsolved claim in number theory. It's
still a statement: it has a definite truth value (mathematicians widely
believe it's true), even though nobody has proved which value that is yet.
"Nobody has settled it" and "it doesn't have a truth value" are different
things, and confusing them is a common early mistake.
Where a statement about n is true
Drag the probe point across the line and read where the open sentence 'n is even' becomes a true statement — at every point it lands on, or only some?
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Worked example
Statement, open sentence, or neither?
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1
Classify: "12 is divisible by 4."
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2
Check whether it has a fixed, checkable truth value: 12 and 4 are both specific numbers, so "divisible by" between them is either true or false right now, with nothing left open.
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3
12 \div 4 = 3, an integer, so the sentence is true. It's a statement — specifically, a true one.
Worked example
Statement, open sentence, or neither?
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1
Classify: "n is divisible by 4."
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2
The structure looks identical to the last example, but n has no fixed value here — nothing tells us which number "n" refers to.
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3
Because the truth of the sentence changes depending on what n is (true at n=12, false at n=13), it has no single fixed truth value of its own. It's an open sentence, not a statement.
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Where this goes next
Computer science
A function's precondition in a program — "the input array is non-empty" — is exactly a statement in this sense: for any specific array handed to the function, it's definitely true or definitely false, never ambiguous. A loosely written specification that leaves a term undefined ("input is reasonable") is really an open sentence disguised as a rule, and it's the single most common source of a bug that "shouldn't have compiled" but did. Chapters 2 and 3 give you the exact tools — connectives and quantifiers — to write these conditions so precisely that a computer can check them.
Chapter 2 gives statements a symbol and a name — propositions — and starts combining them with and, or, and not to build more complex claims out of simple ones.
Check your understanding (counts toward your progress)
Which pair correctly matches each item to statement or open sentence? "(a) 15 is odd. (b) n is odd."
Right on try — credit. Right — the answer had already been shown, so no credit this time. Next try is worth Answer shown after three tries — this one no longer earns credit. You chose to see the answer — this one earns no credit.
Why
15 is a specific number, so (a) has a fixed truth value; n is unspecified, so (b) does not.
Goldbach's Conjecture ("every even integer greater than 2 is the sum of two primes") is unproven. Is it a statement?
Right on try — credit. Right — the answer had already been shown, so no credit this time. Next try is worth Answer shown after three tries — this one no longer earns credit. You chose to see the answer — this one earns no credit.
Not knowing the truth value and not HAVING a truth value are different things -- the claim is either true or false, we just have not settled which.
There's no unfixed variable here -- "every even integer" is a complete, self-contained claim.
Why
A statement only needs a definite truth value to exist, not for that value to already be known.
"n is a prime number" is best classified as:
Right on try — credit. Right — the answer had already been shown, so no credit this time. Next try is worth Answer shown after three tries — this one no longer earns credit. You chose to see the answer — this one earns no credit.
Its truth isn't settled yet -- it depends on what n is.
Same issue -- n has no fixed value here, so there's nothing to call false (or true) yet.
Why
Because n is unspecified, the sentence's truth changes depending on n -- the definition of an open sentence.
Which phrase turns a sentence into an open sentence rather than a statement?
Right on try — credit. Right — the answer had already been shown, so no credit this time. Next try is worth Answer shown after three tries — this one no longer earns credit. You chose to see the answer — this one earns no credit.
Why
An unspecified variable is exactly what leaves a sentence's truth unsettled -- the definition of an open sentence.