1. Nathan Brunelle Department of Computer Science University of Virginia www.cs.virginia.edu/~njb2b/theory Theory of Computation CS3102 – Spring 2014 A tale of computers, math, problem solving, life, love and tragic death

2. Today: Formal Logic Review of discrete math Notation will be used heavily Unambiguous All answers require argument Goals: Develop intuition Practice fluency Note: today we’ll be more formal than I expect on the problems

3. Propositional Logic

4. First Order Logic

7. First Order Logic Inference

8. First Order Logic

10. General Proof Guidelines No need to be super formal Shorter is always better! “A picture is worth 1000 words” Don’t try to surprise people Consider all corner cases

11. Set Theory

12. Notation: {}: Set notation {1, 2, 3} is a set {1, 2, 3, …} is the set of all positive whole numbers

13. Set Theory

20. Useful Sets

21. Problem: (1/4) + (1/4) 2 + (1/4) 3 + (1/4) 4 + … = ? Find a short, geometric, induction-free proof. 1 1

22. Problem: (1/8) + (1/8) 2 + (1/8) 3 + (1/8) 4 + …= ? Find a short, geometric, induction-free proof.

23. Binary Relations