Module 7
Must Be True vs Could Be True.
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.
- must be true — true in every case allowed by the information. One counterexample kills a choice. （必ず成り立つ）
- could be true — true in at least one case. One valid example is enough to keep it. （成り立ちうる）
- cannot be true — true in no case at all. （成り立たない）
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.
日本人がハマる理由
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。
練習問題
難易度は出題者による目安です。College Board の判定ではありません。
If x² = 25, which of the following must be true?
Correct Answer: A
Explanation: x² = 25 allows x = 5 and x = −5. The only statement true in both cases is |x| = 5.
- (C) x = 5: Could be true, but
x = −5is a counterexample. This is the answer to a question that was not asked. - (B) x = −5: Same problem in the other direction;
x = 5breaks it. - (D) x > 0: Broken by
x = −5.
Difficulty Justification
Rated Medium. Choice C 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: D
Explanation: n must be a positive integer below 5: that is 1, 2, 3 or 4. Only n = 4 appears.
- (A) n = 0: Zero is not positive. （0は正の整数ではありません）
- (B) n = 5: "Less than 5" excludes 5 itself — 未満, not 以下. See Module 4.
- (C) n = 4.5: Satisfies the inequality but is not an integer. Both conditions must hold.
Difficulty Justification
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: C
Explanation: A product is zero exactly when at least one factor is zero. "Or" is the correct connective.
- (A) a = 0: Broken by
a = 3, b = 0. - (B) b = 0: Broken by
a = 0, b = 3. - (D) a = 0 and b = 0: Broken by the same examples. "And" demands both, which is far stronger than the information supports.
Difficulty Justification
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.