TMUA Official 2020 Paper 2 · Question 8

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|\leq|a|+|b|

EThe statement

(*)

is false, and the student's proof is fully correct.

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