Week 1 - Sets, Proofs, Logic

Question 1

How do you do a direct proof?

Answer 1

Restate the theorem in the form if(conditions) are true, then (conclusion) is true. Take some notes on what it means for the conclusion to be true and see where it leads. Argue in the direction of the conclusion. Always make sure to start from conditions and argue to the conclusions

Question 2

What is the pigeonhole principle?

Answer 2

If you have more pigeons than holes, then there must be more than one pigeon in a hole

Question 3

What is a constructive proof? What is a non-constructive proof?

Answer 3

A method of proof that demonstrates the existence of a mathematical object by creating or providing a method for creating the object. A non-constructive proof provides the existence of a particular kind of object without providing an example

Question 4

What is a power set?

Answer 4

Power set P(A) of set A means that it is the set of all subsets of A

Question 5

What is DeMorgan’s laws (logic version)?

Answer 5

(~p) V (~Q) is equal to ¬(P∧Q), and (¬P)∧(¬Q) is logically equivalent to ¬(P∨Q).

Question 6

What does contrapositive mean?

Answer 6

Converse and inverse at the same time. Meaning you switch the logic statements, and you negate them.