Module 7
必ず成り立つ vs 成り立ちうる （Kanarazu Naritatsu vs Naritachiuru）
帰国子女・インターナショナルスクール生のための、デジタルSAT 数学の無料教材。独学で進められます。
These two phrasings ask opposite questions, and both appear constantly.
Method for "must be true". Do not try to prove the right answer. Try to break each choice with a single counterexample. Whatever you cannot break is the answer — this is faster and far more reliable than reasoning forwards.
英語は「必ず成り立つのか、成り立ちうるのか、成り立たないのか」という論理の区別を、must・could・cannot という助動詞1語に押し込みます。文の途中に置かれる短い語で、音読しても強く読まれません。日本語は同じ区別を （必ず / 〜する可能性がある / 決して〜ない） のように、副詞や文末の述語という、はるかに長く目立つ形で表します。英語ではこの1語が小さいので読み飛ばしやすく、読み飛ばせば問われている内容が正反対になります。設問を読んだら、must be true か could be true か cannot be true かを先に確定させてから選択肢に入ってください。
If x² = 25, which of the following must be true?
Correct Answer: C
Explanation: x² = 25 allows x = 5 and x = −5. The only statement true in both cases is |x| = 5.
x = −5 is a counterexample. This is the answer to a question that was not asked.x = 5 breaks it.x = −5.Rated Medium. Choice A is correct under "could be true" and wrong under "must be true" — the word alone decides it.
If n is a positive integer and n < 5, which of the following could be true?
Correct Answer: C
Explanation: n must be a positive integer below 5: that is 1, 2, 3 or 4. Only n = 4 appears.
Rated Easy. Each distractor violates exactly one of the three stated conditions — positive, integer, less than 5 — so the question rewards checking every condition separately.
If ab = 0, which of the following must be true?
Correct Answer: D
Explanation: A product is zero exactly when at least one factor is zero. "Or" is the correct connective.
a = 3, b = 0.a = 0, b = 3.Rated Hard: it turns entirely on and versus or. Note that mathematical "or" is inclusive — it permits both being zero, but does not require it.