Week 7: The End of Proof techniques and onto Proofs of Complexity

      We are finally done the proof techniques section of the course and have moved on to complexity proofs. I feel like this section is the real major point of taking this course for computer science. Even though all of the previous sections are essential to know before learning this section, they feel like they are only tools to work with for this upcoming section. However we haven't really seen too much yet, however I suspect we will be learning more about complexity proofs next week.

How I currently feel about Proof Structures and a Summary of Proofs we learned so far

    It has definitely been a challenging section in the course since we took our knowledge of logical notation and have learned how to form proof structures using our understanding of logical notation when a for all symbol is introduced we see it as an opponent picking a variable within the specified domain that we have know control over. However when there is an existential quantifier introduced we pick the variable from a specified domain. I find this is a great analogy to learn in order to understand the differences of these quantifiers when forming a proof.

Since not much has happened this week I will just go over the many different techniques we learned in order to prove some claim such as:

• Proof by Contradiction
• Direct Proof
• An Indirect proof (e.i. using the contrapositive)
• Also using other proven claims and theorems to prove other claims (e.i. Number Theory, Definitions of odd and even, and also Definitions of ceiling and floor of a real number )