Last Update: 11 October 2010
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Index to all lecture notes
…Previous lecture
Strategy & Proof:
1. | Odd(c) | : temporary assumption to show Even(c+1) by Direct Proof |
2. | ∃k[c=2k+1] | : 1; def of Odd |
3. | c=2k_{1}+1 | : 2; EI |
4. | c+1=2k_{1}+1+1 | : 3; algebra |
5. | c+1=2k_{1}+2 | : 4; algebra |
6. | c+1=2(k_{1}+1) | : 5; algebra |
7. | ∃j[c+1=2j] | : 6; EG |
8. | Even(c+1) | : 7; def of Even |
9. | Odd(c) → Even(c+1) | : 1–8; Direct Proof |
10. | ∀x[Odd(x) → Even(x+1)] | : 9; UG |
Strategy & proof:
But might be easier to show the contrapositive: (¬Even(c) → ¬Odd(7c–5)):
∴ supp. ¬Even(c), & show ¬Odd(7c–5):
But now we can use another strategy that I didn't mention:
i.e., instead of supposing ¬Even(c), let's supp.
Odd(c),
which is mathematically equivalent (you should know that!)
and instead of trying to show ¬Odd(7c–5),
let's try to show Even(7c–5)
which is mathematically equivalent.
So: supp. ∃k[c=2k+1]
& show ∃j[7c–5 = 2j]:
i.e., find j such that 7c–5 = 2j:
∴ 7c–5 = 7(2k+1)–5
= 14k+7–5
= 14k+2
= 2(7k+1)
∴ take j = 7k+1.
QED
So we need to show: ¬Q(√2)
∴ 2 = a^{2}/b^{2}
∴ 2b^{2} = a^{2}
∴ Even(a^{2})
∴ Even(a), by lemma (Rosen, p.85: 16)
…to be continued…