make sure your deductive reasoning flows logically from one statement to another

Question

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 q

Explanation / Answer

Question1. (a)

Jane wins Math | True (given already) - (1)

Jane wins Chemistry | False , since jane wins Math cannot possible win both - (2)

Pete wins Math | False, since Jane wins Math, and both can't win Math - (3)

Pete wins Chemistry | True (From (1), (2), (3) )

(b)

Let 1 represent truth and 0 represent false.

Truth table:-

John | Bill | Sam Result

0 0 0 Case not possible, as either john or bill is telling truth

0 0 1 Case not possible, as either john or bill is telling truth

0 1 0 Case possible, Sam is lying , john is lying

0 1 1 Case not possible, as either sam or bill is lying, not both are truthful

1 0 0 Case not possible, as either sam or bill is lying, not both are lying

1 0 1 Case possible, here john is truthful and sam as well

1 1 0   Case not possible, as either john or bill is telling truth, not both

1 1 1   Case not possible, as either john or bill is telling truth, not both

From the 2 cases, the inference that either john is telling the truth or sam is lying is false.

(c) Let good sales be represented by 1 and high expenses be 1 , bad sales be 0, bad expense be 0

Sales | Expense | Result

0 0 Case not possible, as with down expense, boss will be happy, and unhappy with 0 sales,

0 1 Case possible, as boss is unhappy here with low sale, high expense

1 0 Case possible, as boss is happy with high sale and low expense

1 1 Case not possible, as boss will be happy with good sales, and unhappy with high expense

Clearly, both expense and sales can't go up simultaneously.

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