Honors Calculus I - 1

Between calculus and math analysics

axiom, proposition, lemma, theorem(most important), corollary (direct consequences)

Mathematical Proof

Direct proof
Induction : Regular & Strong

Regular use the previous one step, while strong induciton use all the previous steps(like recursive)
Contraposition (negate the hypothesis)

for p => q, we let !q => !p
1. We prove the statement by contraposition
2. Suppose…
3. Then…
4. Therefore, …, by contraposition
origin : p => q

reverse : q=> p

inverse: -p => -q

contrapositive -q => -p

The origin an contrapositive are logical equivalent, while reverse and inverse are equivalent

The not statement is -p
Contradiction (logical contradiction)

Use !q to get a contradition
Exhaustion
Counterexample

To prove $A=B$ , we can prove $A \subset B$ and $B \subset A$