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==== Direct proof: for every real number, there's another number that's even closer to zero. - How to write as a formula? - How to prove it (direct proof)? Any glitches? In lecture: If n is odd, then n^2 is odd. Note that we need a def'n of odd. ==== Another proof, in gory detail (notes only): What is the definition of a number being even? Odd? Show: if n is even, then n+1 is odd. 1. n is even premise (NB: n is arbitrary) 2. ∃k.(n=2k) def'n even 3. n=2k existential instantiation (NB k is not arbitrary) ... 15. ¬∃z.(n+1=2z) ... Proof by contradition: suppose n+1=2z, for some integer z. Then 2k+1=2z z=k+1/2 z-k=1/2 Contradiction. (since no integers differ by less than 1 -- Th'm) Thus 16. ¬(n+1 is even) by line15 + def'n of even 17. n+1 is odd by line16 + def'n of odd We can re-write this as a direct proof: Show: if n is even, then n+1 is odd. 1. n is even premise (NB: n is arbitrary) 2. ∃k.(n=2k) def'n even 3. n=2k existential instantiation (NB k is not arbitrary) 4. ∀z.∀y.(z+1/2 &neq; y) th'm: no integers differ by less than 1. 5. ∀y.(k+1/2 &neq; y) universal instantiaion[z/k], line 4, 4. ¬∃m.(k+1/2 = m) by def'n of integer; (m arbitrary) 5. ¬∃m.(2k+1 = 2m) line4 + multiply equals by equals 6. ¬∃m.( n+1 = 2m) line 3 + subst equals for equals. 7. ¬(n+1 is even) by line15 + def'n of even 8. n+1 is odd by line16 + def'n of odd Strategies - Use your definitions, to go a step or two forwards. - Write down your goal. Use def'ns, to go a step or two backwards. (This often helps you solidifies the meaning of the definitions for you.) - Try a direct proof, if you can come up with a constructive algorithm; try an indirect proof if you are trying to show how you think *any* constructive algorithm must fail. (This is the same thought process used in designing an algorithm, and in debugging: Figure out how your algorithm will (eg) search through *any* list-of-names. To debug, come up with the smallest example which the algorithm won't work for. - Another strategy: see if you can show the contrapositive. Example: If m^2 is even, then m is even. To show "a iff b": - show "if a, then b" - show "if b, then a" Example: A number is rational iff its decimal expression repeats. Also, "TFAE": - x is irrational - the decimal expansion of x never repeats - 3x is irrational.
©2008, Ian Barland, Radford University Last modified 2008.Sep.26 (Fri) |
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