Week 5: Further Proofs

This week's lecture included our midterm. After the midterm, we learned some additional methods of proving.

Last week, we learned how to prove a universally quantified implication- it was simple and straightforward. This week, the first thing we learned was using contrapositive to prove an implication. For example, to prove P implies Q, we can prove not Q implies not P; it achieves the same result. We use this when the reverse direction is easier to prove than the original.

Next up, we learned the usage of contradiction- a special case of contrapositive. This is a tricky one, and it involves assuming not Q instead of Q. Using this method, we assume P implies not Q, and look for a contradiction and disproving it.

For proof of an existential, we just need to find one example that satisfies.

The overall summary of the lecture is that one proof may consist of many different structures, and that with multiple quantifiers, zoom into smaller parts of the statement.