Question 1

make sure your deductive reasoning flows logically from one statement to another. You should edit and revise your proofs to make sure they are clear and concise. Typesetting your solutions in BTeX(www.latex-project.org) is pre- ferred (and streamlines revisions), but you may also submit written solutions. The ISTEXsource for our assignments is posted online. This assignment does not have any (ungraded) warm-up questions, but most will. Question 1. Analyze the following deductive arguments and analyze their logical forms, specifying the premises and conclusions in propositional logic. Use truth tables to arue whether the reasoning is valid or invalid (a) Jane and Pete won't both win the math prize. Pete will win either the math prize or the chemistry prize. Jane will win the math prize. Therefore, Pete will win the chemistry prize. Either John or Bill is telling the truth. Either Sam or Bill s lying. Therefore, either John is telling the truth or Sam is lying. (c) If sales go up then the boss will be happy. If expenses go up then the boss won't be happy. Therefore, sales and expenses will not both go up. Question 2. Demonstrate the following equivalences and tautologies. (a) Use truth tables to show that P Q is equivalent to (FAQ)v (-P Q) (b) Use logical deductions (ie., apply the Laws) to show that P Q) V (P R) is equivalent to P (QV R) (c) Use a method of your choice to show that (PQ) v (Q R) is a tautology VLSI company, and your first task is to construct two combinatorial r of OR gates, AND Question 3. You recently started working in circuits that produce from input bits p, g,r certain desired outputs. You can use any numbe gates, and inverters. The desired outputs are nstructin p) r Do you observe any issues with this combinatorial circuit? Is it worth co p V q at all? (a) p