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# Math Math 015

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This 7 page Class Notes was uploaded by Sindhu Furtari on Monday September 19, 2016. The Class Notes belongs to Math 015 at University of Maryland taught by Ben Bezejouh in Fall 2016. Since its upload, it has received 3 views.

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Date Created: 09/19/16

CMSC 250 Fall 2016 Homework #1 Posted: 09-06-2016 Due: 09-13-2016 Student’s first and last name:enjamin Mao Grade (grader only): Student’s UID: 114042217 Student’sSection: 0303 University Honor Pledge: I pledge on my honor that I have not given or received anyunauthorized assistanceonthisassignment/examination. Print the text of the University Honor Pledge below : I pledge on my honor that I have not given or received any unauthorized assistance on this assignment /examination. Signature: BenjaminMao Problem (1): Propositional Logic (50 pts) Question (a): Maximal Simplifications (30 pts) Using the axioms of propositional logic equivalence outlined in lecture, simplify the following compound statements as much as possible . Recall the approach we used in classfor every derivation you make, write down the axiom you apply on the right of the derivation. (i) (k ∧ l) ∨ (k ∧ m) ∧ k ∧ (l ∨ m) (10 pts) (ii) z ∧(q ∨( w ∧q)) ∧( z ∨ q) (20 pts) BEGIN YOUR ANSWER TO QUESTION (A) BELOW THIS LINE (k ∧ l) ∨ (k ∧ m) ∧ k ∧ (l ∨ m) –Given ????∧(l∨m)∧k∧(l∨ ∼m) –Distribution ????∧(l∨m)∧(l∨ ∼m) – Tautology ????∧l∨(m∨ m) – Distribution ????∧l – Law of Non -contradiction 1 09-13-2016 – CMSC 250 2 CONTINUE YOUR ANSWER TO QUESTION (A) BELOW THIS LINE z ∧(q ∨( w ∧q)) ∧( z ∨ q) –Given ????∧( ((q∨~????) ∧(q∨q) ∧~(????∨q) – Distributive and De Morgan’s law ????∧ (q∨~????) ∧~(????∨q) - Tautology (????∧ q)∨(z∧ ~????) ∧~(????∨q) - Distribution (z∧~????) ∧(q∧ ~????) )∨(z∧ ~????)-Association (z∧ ~????)- Law of Non -contradiction 09-13-2016 – CMSC 250 3 Question (b): Equivalences (5 pts each for 20 pts total) The statements (i) -(iv) contain the implicatio⇒) and biconditional ( ⇔) operators. Using known equivalences discussed in lecture, rewrite thesuch that they contain neither operator! Then, maximally simplify them , so that they contain the smallest possible number of terms. (i) ( p ∧ q) ⇒ r (ii) ( p ∨ q) ⇒ (r ∨ q)∼ (iii) ( p ⇒ r) ⇔ (q ⇒ r) (iv) (p ⇒ (q ⇒ r)) ⇔ ((p ∧ q) ⇒ r) BEGIN YOUR ANSWER TO QUESTION (B) BELOW THIS LINE (p ∧ q) ⇒ r –Given p= (p ∧ q), q=r –Substitution p⇒q ∼(p ∧ q)∨r –Definition of Material Implication p∨ q∨r –De Morgan ’ s law ( p ∨ q) ⇒ (r ∨ q) -Given p= ( p ∨ q), q= (r ∨ q) –Substitution p⇒q ∼ ∼ ∼ ( p∨ q)∨ (r ∨ q) –By Definition (p∧ ∼ q)∨ (r ∨ q) –Double Negation and De Morgan’s (p∨r)∧( ∼q∨ q) –Association (p∨r)∧ ∼q -Tautology (p ⇒ r) ⇔ (q ⇒ r) –Given ∼ ∼ p∨r∧ q∨r –Definition r∨ ∼(p ∨q) –Distribution and De Morgan ’ s (p ⇒ (q ⇒ r)) ⇔ ((p ∧ q) –Given ∼p∨( q∨r)∧ ∼( p∧ q) ∨p –Definition ∼ ∼ ∼ ∼ p∨( q∨r)∧ p∨ q∨p –De Morgan ’ s ∼p∨( q∨r)∧ ∼ q –Law of Excluded M iddle s (∼p∨ ∼q)∧( ∼p∨r)∧ ∼q –Distribution r –Double Tautology 09-13-2016 – CMSC 250 4 CONTINUE YOUR ANSWER TO QUESTION (B) BELOW THIS LINE 09-13-2016 – CMSC 250 5 Problem (2): Logic in“real ” life (30 pts) It was a quiet day in the land of HonestDishonest Llamas, inhabited by lla-like creatures wheither always tell the truth or constantly lie about everything. On such a fateful day, Carl the llama was in search of a butch er shop to purchase some meat for his meat dragon. On the road towards the nearby town, i-waythree intersection, he encountered two other llamas, Paul and Sam. The following dialogue ensued: – Carl: Hey guys, I’m looking to purchase some meat for my meat dragon. Do I need to go left or rreachto the local butcher? – Paul: Caaaaaaaaaaaaaarl! The nearest butcher is downtown, or the right road leads to him, Carl! – Sam: Caaaaaaaaaaaaarl! The butcher is downtown, and the road to the right leads to him, Carl! – Paul: Sam’s lying Carl, please don’t believehim! – Sam: *shrugs* If the butcher’s downtown, then the road to the right leads to him, Carl! Carl then took a minute to think about what he should do, and decided to take the road toDid Ca rl. make the correct decision, and why? Note that, for full credit, your answer must contain an explanation based on a truth table! BEGIN YOUR ANSWER TO PROBLEM (2) BELOW THIS LINE p q p∨q p∧q p⇒ q ∼p∨ q∼ T T T T T F F F F F T T p=Butcher downtown q=Take the right road Both Sam and Paul are lying and therefore Carl was right not to listen to them and take the left road instead. If Paul were telling the truth, the butcher would be downtown or Carl would have to take the right road to get there. But Paul also says Sam is lying at the same time, which is incorrect. Since Paul can only tell truths or lies and he has told both a truth and a lie, he must be lying about Sam lying and the road leading to the butcher shop being downtown or on the right. Sam is a bit more involved since he changes his story. If Sam were telling the truth, both his statements would be true and hence he would not feel the need to change the first one. Therefore, he must be lying. If he were lying, his first statement would be false, but his second one would be true. This wouldn’t be possible since he is obligated to tell either the truth and nothing but the truth or lie consistently. Like the case with Paul, he cannot tell both one truth and one lie. Therefore, he must also be lying. Since both of the llamas are lying, Carl was correct in going to the left. He will find the butcher there for his meat dragon. 09-13-2016 – CMSC 250 6 CONTINUE YOUR ANSWER TO PROBLEM (3) BELOW THIS LINE 09-13-2016 – CMSC 250 7 Problem (3): Valid reasoning in real life (2 pts each for 20 pts total) In the following real -life arguments, write down the name of the law of reasoning that you believe occurs, whether you agree with the statements made or not: Modus Ponens, Modus Tollens, Disjunctive / Conjunctive addition, Conjunctive Simplification, Disjunctive / Hypothetical syllogism. (i) Since I have both oranges and apples, it’s true that I have apples. Conjunctive Simplification Modus Tollens (ii) If I studied, I would have passed 250. However, I did not pass, which means that I did not study. Modus Ponens (iii) All men are mortal. Socrates is a man. Therefore, Socrates is mortal. (iv) All smart people can become developers, and developers make good money. Therefore, all smart people make Hypothetical Syllogism good money. (v) Well, you wanted either a Boston Cr` eme donut or an apple fritter, and they were out of apple fritters. So here’s your donut. Disjunctive Syllogism (vi) Harry is my dog, and all dogs love bones. So I’m expecting him to love thisbone. Hypothetical Syllogism (vii)I have experience in Android development, so I can definitely respond to this job ad that needs experienced Android or iPhone developers. Disjunctive Addition (viii) With the appropriate talent, drawing becomes easy. Since I find drawing as hard as I do, it’s clear that I’m untalented. Modus Tollens (ix) Jason is Greek and Greek people eat a lot of red meat. It follows that Jason eats a lot of red meat. Hypothetical Syllogism m (x) Look, it was either that job or your mental health. Since you were fired from the job, you can at least look towards you r health. Disjunctive Syllogism

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