C241 PLTL SESSION – 2/10/2015 More Proofs & More Graphs Warm-Up Exercise • Grab a worksheet • Begin completing the worksheet in pairs or small groups Problem 1 Using a proof by contraposition, prove that for any quadratic equation, if the values given by the quadratic formula are real, then that equation crosses the x-axis. Problem 2 Draw the following situation as a graph: Becky likes Jason & Alex, Chris likes Alex & Becky, Alex likes everyone, and Jason hates everyone, (including himself). Apart from Jason, you may assume that everyone likes themselves. Problem 3 Draw the following situation as a graph, (like the one from class today): In the domain of all people, everyone is mortal. Alex is nice and he trusts everyone, but no one trusts Alex. Ben and Chris trust each other, but only Ben is nice. Also, assume that everyone trusts themselves. (Note: In this context, “everyone” means Alex, Ben, and Chris. Also, think of the truth relation as a two-way street). T(x, y) = x trusts y M(x) = x is mortal N(x) = x is nice Problem 4 Given the situation drawn above, translate the following into statements of predicate logic, and then assess the truth of each statement: • A. Everyone is mortal. • B. Everyone is mortal or nice. • C. Everyone trusts someone. • D. Everyone who is trustworthy is also nice. • E. Someone trusts everyone. • F. Everyone trusts someone nice, excluding themselves. Problem 5 Prove the following claim (by contradiction): There are infinitely many primes. Quantifier Group Work • On a spare piece of paper, please write down any questions or issues that you have with quantifiers. • In different groups than those you did your worksheets in, discuss the difficulties you’ve been having with quantifiers. • Find a problem from the notes or the textbook that you feel addresses your issue and begin completing this problem in your group.