Proof and logic

Sunday 24 October 2004 at 3:15 pm | In Articles | 3 Comments

Mathematical proof is one of the topics that students find very difficult. Many of them assume what they are trying to prove, end up with a true statement and then think they have proved the result. Studying truth tables, particularly the implication operator may well help. See Logical Operations and Truth Tables

On a mathematics forum a student wanted to use its \LaTeX facilities (provided by LatexRender of course 8-) ) to help a friend show how to prove

    \sin{(x+\frac{\pi}{4})}+\cos{(x+\frac{\pi}{4})}=\sqrt{2}\cos{x}

This is the original ‘proof’ they gave. Although they have now changed it I get the impression that I failed to convince them of the faulty logic; how would you explain what is wrong?

    \sin{(x+\frac{\pi}{4})}+\cos{(x+\frac{\pi}{4})}=\sqrt{2}\cos{x}

    For this problem you need to know the addition formulas:
    \
\sin{(x+t)}=\sin{x}\cos{t}+\cos{x}\sin{t}\
\cos{(x+t)}=\cos{x}\cos{t}-\sin{x}\sin{t}

    Using these formulas in the problem we can turn it into:
    \
(\sin{x}\cos{\frac{\pi}{4}}+\cos{x}\sin{\frac{\pi}{4}})+(\cos{x}\cos{\frac{\pi}{4}}-\sin{x}\sin{\frac{\pi}{4}})=\sqrt{2}\cos{x}

    Then we use the fact that \sin{\frac{\pi}{4}}=\frac{\sqrt{2}}{2} and \cos{\frac{\pi}{4}}=\frac{\sqrt{2}}{2}

    This changes it to:
    \
\cancel{\frac{\sqrt{2}}{2}\sin{x}}}+\frac{\sqrt{2}}{2}\cos{x}+\frac{\sqrt{2}}{2}\cos{x}-\cancel{\frac{\sqrt{2}}{2}\sin{x}}=\sqrt{2}\cos{x}

    Here we cancelled out the terms that equals zero and then adding together what we have we end up with:

    \sqrt{2}\cos{x}=\sqrt{2}\cos{x}

3 Comments »

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  1. I don’t get why that is ‘faulty logic’…
    If the writer had expressed the same steps by beginning with
    sin(x + pi/4) + cos(x + pi/4)

    and without writing the ‘= sqrt(2) cos x’ part at each stage, it would clearly have been an acceptable demonstration of equality.

    Alternatively we could take
    ‘= sqrt(2) cosx’
    in this instance to mean
    ‘we wish to show this equals sqrt(2) cosx’

    While we might pick at the notation, the logic of what is going on is fairly sound, in my opinion (unless I have missed something?)

    Comment by Ronald — Sunday 14 November 2004 3:43 pm #

  2. It is not acceptable to start with what you want to prove and then end up with a true statement. You must start with what you know and end up with what you want to prove.
    Otherwise you’d get nonsense like
    -1=1 so (-1)2=12 and thus 1=1 which is true.

    In this case the mathematics is all there but the student should have started with one side and worked towards the other or else (but more difficult to write) show that every step is reversible. Starting with \sqrt{2}\cos x=\sqrt{2}\cos x and ending with the final result is a proof, but that’s not what they wrote.

    Comment by Steve — Sunday 14 November 2004 4:04 pm #

  3. For some reason my e-mail address below is wrong.

    You’re right, of course, but in this case the RHS remains unchanged throughout – so they *have* started from one side and worked towards the other side.

    Comment by Ronald — Thursday 23 December 2004 12:17 pm #

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