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QLDSpecialist MathematicsQuick questions

Unit 3: Mathematical induction, and further vectors, matrices and complex numbers

Quick questions on Trigonometric proofs and methods of proof in QCE Specialist Mathematics Unit 3

3short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.

What is direct proof?
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A direct proof of "if PP then QQ" assumes PP and derives QQ through a sequence of valid steps. For example, to prove that the product of two even integers is even, write m=2am = 2a and n=2bn = 2b for integers a,ba, b, so mn=4ab=2(2ab)mn = 4ab = 2(2ab), which is even because 2ab2ab is an integer. Every step must follow from a definition or an earlier line.
What is proof by contrapositive?
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The statement "if PP then QQ" is logically equivalent to its contrapositive "if not QQ then not PP". When the negation of QQ is easier to work with, prove the contrapositive instead. To prove "if n2n^2 is even then nn is even", the contrapositive "if nn is odd then n2n^2 is odd" is direct: n=2k+1n = 2k+1 gives n2=4k2+4k+1=2(2k2+2k)+1n^2 = 4k^2 + 4k + 1 = 2(2k^2 + 2k) + 1, which is odd.
What is proof by contradiction?
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Assume the statement is false and derive a contradiction. The classic example is the irrationality of 2\sqrt2: assume 2=pq\sqrt2 = \frac{p}{q} in lowest terms, so p2=2q2p^2 = 2q^2. Then p2p^2 is even, so pp is even, say p=2rp = 2r, giving 4r2=2q24r^2 = 2q^2, hence q2=2r2q^2 = 2r^2, so qq is also even. But then pp and qq share the factor 22, contradicting "lowest terms".

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