Catch-up Session 2
Methods of Proof
- Counter Example Method
- Prove that a statement is false by use of a counter-statement.
- Universal Proof
- Applies to all integers e.g. odd if $a=2k+1$
Example
\((a+b)^2=a^2+b^2\)
Holds for:
- Some integers
- All integers
- No integers
The answer is that it holds for some integers as when $a=0,\ b=0$. \((0+0)^2=0^2+0^2\) This also follows for $a\neq0,\ b=0$.
The two proofs I gave as answers to the question count for both a counter example proof and a universal proof.
Example of Proof by Contradiction
Suppose for a proof by contradiction that $P_n$ is the largest prime number. Therefore, $P_1\ldots P_n$ are all the primes.
Consider $P=P_1\times P_2\ldots P_n+1$
Case 1: $P_p$ is prime
Case 2: $P_p$ not prime. Then $P$ must have prime member.