Flashcards
Proof
Cards in this deck (11)
- 1With constructive proof, we need to....Manipulate one side of a problem to give the other (e.g. trig identity questions)
- 2With exhaustive proof, we need to....Try every possible situation and check it's right
- 3To use a counter example, we need to....Find a single example that proves the statement is not true
- 4We can represent even numbers by....2n, 2n-2, 2n+2, etc
- 5We can represent odd numbers by....2n+1, 2n+3, 2n-1, etc
- 6We can represent consecutive numbers by....n, n+1, n+2, n+3, etc
- 7We can represent multiples of (5) by....5n, 5n+5, 5n-10, etc
- 8When we use 2 number which aren't consective, we need to make sure we use.....Different letters (so n and m, not just n)
- 9What is the method to use proof by contradiction to prove there are an infinite number of prime numbers?1. Assume the opposite (they're aren't an infinite number)
2. Make a obvious statement showing this; 2,3..... pn are primes, where pn is the largest prime
3. Do something to this to prove the original assumption wrong and summarize: Let M=2x3x5....x pn +1. So 2x3x5...x pn is divisible by all primes but M (2x3x5x... +1) is not, and therefore is prime. M>pn and therefore contradicts the initial assumption, and therefore there are an infinite number of primes - 10What is the method to use proof by contradiction to prove there root(2) is irrational?1. Assume root(2) is rational
2. Make a obvious statement showing this; root(2) = a/b where a/b in in simplest form
3. Do something to this to prove original assumption wrong and summarize: Square both sides (2=a^2/b^2 , rearrange to get 2b^2 = a^2 , and therefore a^2 must be even, and so must a, so we can say a=2n). We can repeat this for b by rearranging 2=4n^2/b^2 to find that b must also be even, so b=2m. Finally we say that root(2)=a/b=2n/2m and therefore a/b is not lowest terms, contradicting our original assumption therefore root(2) is irrational. - 11How would we prove everything else using proof by contradiction?Apply these 2 methods for their appropriate questions.