A student is asked to prove whether the following statement (×) is true or false:
(×) For all real numbers a and b, |a + b| < |a| + |b|
The student's proof is as follows:
Statement (×) is false. A counterexample is a = 3, b = 4, as |3+4| = 7 and |3| + |4| = 7, but 7 < 7 is false.
Which of the following best describes the student's proof?
- AThe statement (×) is true, and the student's proof is not correct.
- BThe statement (×) is false, but the student's proof is not correct: the counterexample is not valid.
- CThe statement (×) is false, but the student's proof is not correct: the student needs to give all the values of a and b where |a+b| < |a| + |b| is false.
- DThe statement (×) is false, but the student's proof is not correct: the student should have instead stated that for all real numbers a and b, |a+b| ≤ |a| + |b|.
- EThe statement (×) is false, and the student's proof is fully correct.
Show the answer and worked solution
answer · E
- AThe statement (×) is true, and the student's proof is not correct.
- BThe statement (×) is false, but the student's proof is not correct: the counterexample is not valid.
- CThe statement (×) is false, but the student's proof is not correct: the student needs to give all the values of a and b where |a+b| < |a| + |b| is false.
- DThe statement (×) is false, but the student's proof is not correct: the student should have instead stated that for all real numbers a and b, |a+b| ≤ |a| + |b|.
- EThe statement (×) is false, and the student's proof is fully correct.
To disprove a "for all" statement you need exactly one pair of values that makes it fail, and the student has one: with a = 3 and b = 4 both sides are 7, so the strict inequality |a+b| < |a| + |b| does not hold. That is all a disproof requires. Listing every failing pair is not needed, and stating the correct inequality |a+b| ≤ |a|+|b| would be a different (true) claim rather than a repair of this proof. So the statement is false and the proof stands.