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# MATH FOR LIB ARTS I MGF 1106

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This 17 page Class Notes was uploaded by Colin Goodwin PhD on Thursday September 17, 2015. The Class Notes belongs to MGF 1106 at Florida State University taught by James Wooland in Fall. Since its upload, it has received 15 views. For similar materials see /class/205663/mgf-1106-florida-state-university in Mathematics (M) at Florida State University.

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Date Created: 09/17/15

PART 2 MODULE 2 THE CONDITIONAL STATEMENT AND ITS VARIATIONS THE CONDITIONAL STATEMENT A conditional statement is a statement of the form quotIf p then q quot The symbol for this quotifthenquot connective is the arrow gt That is the statement quotif p then qquot is denoted p gtq EXAMPLE 221 Let p represent quotYou drink Pepsi quot Let q represent quotYou are happy quot In this case p gtq is the statement quotIf you drink Pepsi then you are happy quot TERMINOLOGY quotYou drink Pepsiquot is called the antecedent quotYou are happyquot is called the consequent More generally the antecedent is associated with the if part of a conditional statement while the consequent is associated with the then part of a conditional statement EXAMPLE 222 Let p be the statement quotIt rainsquot Let q be the statement quotI stay homequot Symbolize each statement 1 If it rains then I stay home 2 It is not the case that if it rains then I stay home 3 If I don39t stay home then it doesn t rain 4 It is not the case that if I stay home then it doesn t rain Solutions to EXAMPLE 222 1 p gtq 2 p gtq Note that in this case it is the entire if then statement rather than just one or both of its components than is being negated 3 q gtp 4 q gtp For any conditional statement there are several other similarsounding conditional statements Some of these variations have special names VARIATIONS ON THE CONDITIONAL STATEMENT Direct statement Converse Inverse Contrapositive If p then q If q then p If not p then not q If not q then not p P gtq q gtP NP gtq Nq gtP EXAMPLES Direct statement If you drink Pepsi then you are happy Converse If you are happy then you drink Pepsi Inverse If you don39t drink Pepsi then you aren39t happy Contrapositive If you aren t happy then you don t drink Pepsi EXAMPLE 223 Symbolize this statement taken from the instructions for IRS From 1040 line 10 If you received a refund of state income taxes or you received a refund of local income taxes then if your itemized deduction of state income taxes resulted in a tax bene t or your itemized deduction of local income taxes resulted in a tax bene t then you must report this tax bene t as income Let p you received a refund of state income taxes q you received a refund of local income taxes r your itemized deduction of state income taxes resulted in a tax benefit s your itemized deduction of local income taxes resulted in a tax benefit w you must report this tax benefit as income World Wide Web Note For practice problems involving translation of statements from words into symbols and vice versa visit the companion website and try THE SYMBOLIZER EXAMPLE 224 Let p be the statement quotYou drink Pepsi quot Let q be the statement quotYou are happy quot Make a truth table for the statement p gtq The solution to the previous example illustrates the following FUNDAMENTAL PRPOERTY OF THE CONDITIONAL STATEMENT The only situation in which a conditional statement is FALSE is when the ANTECEDENT is TRUE while the CONSEQUENT is FALSE EXAMPLE 225 1 Let p represent a true statement while q and r represent false statements Determine the truth value of this compound statement p gtqvr 2 Let p s and w represent true statements while q r and u represent false statements Determine the truth value of this compound statement NP gt qVrAWOD V11VWl Hint for problem 2 This particular problem is not as complicated as it at rst appears to be World Wide Web Note For practice problems involving truth values of symbolic statements visit the companion website and try THE LOGICIZER EXAMPLE 226 Complete the following truth table World Wide Web Note For practice problems involving truth tables visit the companion website home page and try THE TRUTH TABLER A FACT ABOUT EQUIVALENCY quotpr then aquot is logically equivalent to quotnotp or qquot Symbol ically p gtq E p vq We can use a truth table to verify this claim EXAMPLE 227 Select that statement that is logically equivalent to quotIf you don39t carry an umbrella you39ll get soaked quot A You carry an umbrella and you won t get soaked B You carry an umbrella or you get soaked C You don39t carry an umbrella and you get soaked D You don39t carry an umbrella or you get soaked E You leave your umbrella in the classroom so you get soaked anyway THE NEGATION OF THE CONDITIONAL STATEMENT The negation of quotifp then qquot is quotp and not qquot Symbolically p gtq EpAq We can use a truth table to verify this claim EXAMPLE 228 1 Select the statement that is the negation of quotIf you know the password then you can get in quot A If you don39t know the password then you can get in B You don39t know the password or you can get in C You don39t know the password and you can39t get in D You know the password and you can39t get in 2 Select the statement that is logically equivalent to quotIf you pass MGF1106 then a liberal studies math requirement is fulfilled quot A If a liberal studies math requirement is fulfilled then you passed MGF1106 B You pass MGF1106 and a liberal studies math requirement is fulfilled C You don39t pass MGF1106 or a liberal studies math requirement is fulfilled D You pass MGF1106 or a liberal studies math requirement is not fulfilled 3 Select the statement that is the negation of quotIf you have income from royalties then you must complete Schedule E quot A You have income from royalties and you must complete Schedule E B You have income from royalties and you don39t have to complete Schedule E C You have income from royalties or you must complete Schedule E D You have income from royalties or you don39t have to complete Schedule E EXAMPLE 229 1 Select the statement that is the negation of quotIf we get a pay raise then we will be contentquot A If we don39t get a pay raise then we won t be content B We get a pay raise and we are content C We get a pay raise and we aren t content D We don39t get a pay raise or we aren39t content 2 Select the statement that is logically equivalent to quotIf it is raining then we will watch TV quot A It isn39t raining or we don39t watch TV B It isn39t raining or we watch TV C It is raining and we watch TV D It is raining and we don39t watch TV E It is not safe to watch TV in the rain 3 Select the statement that is the negation of quotIf a dog wags its tail then it won t bite quot A A dog wags its tail and it bites B A dog wags its tail and it doesn39t bite C A dog doesn39t wag its tail or it bites D If a dog doesn39t wag its tail then it will bite SOME INFORMAL EQUIVALENCIES FOR THE CONDITIONAL STATEMENT quotIf p then qquot is equivalent to quotAll 13 are q quot quotIf p then not qquot is equivalent to quotN0 13 are q quot For example quotIf something is a poodle then it is a dogquot is a round about way of saying quotAll poodles are dogsquot Likewise quotIf something is a dog then it isn39t a catquot means the same as quotNo dogs are cats quot EXAMPLE 2210 1 Select the statement that is the converse of quotIf I had a hammer I would hammer in the morning quot A If I didn39t have a hammer I wouldn t hammer in the morning quot B If I don39t hammer in the morning I don39t have a hammer C If I hammer in the morning I have a hammer D If I had a ham I would eat ham in the morning 2 Select the statement that is the inverse of quotIf it rains then I won39t go to classquot A If I don39t go to class then it rains B If it doesn t rain then I will go to class C If I go to class then it isn39t raining D Since it s Friday I probably won t go to class anyway 3 From Shakespeare Henry IV Part 11 Select the statement that is the negation of this line spoken by Falstaff addressing Doll Tearsheet quotIf the cook help sic to make the gluttony you help to make the diseasesquot A If the cook doesn39t help to make the gluttony you don39t help to make the diseases B If you help to make the diseases the cook helps to make the gluttony C If you don39t help to make the diseases the cook doesn t help to make the gluttony D The cook helps to make the gluttony and you don t help to make the diseases FACTS ABOUT CONVERSEINVERSECONTRAPOSITIVE The direct statement is equivalent to the contrapositive Iqu Nq gtP The converse is equivalent to the inverse 01 PE NP gtq However the converse is NOT equivalent to the direct statement and the inverse is NOT equivalent to the direct statement These claims can be verified by using truth tables If you make a truth table having columns for all four statements listed above you will see for instance that the column for p gtq is identical to the column for q gtp but these two columns are different from the column for q gtp and different from the column for p gtq However the column for q gtp will be identical to the column for p gtq EXAMPLE 2211 1 Select the statement that is logically equivalent to quotIf today is Sunday then school is closed quot A If today isn39t Sunday then school isn39t closed B If school is closed then today is Sunday C If school isn39t closed then today isn39t Sunday D A B amp C are all equivalent to the statement above 2 Select the statement that is logically equivalent to quotIf you are a duck then you aren t willing to waltz quot Adapted from Lewis Carrol A If you aren t willing to waltz then you are a duck B If you aren t a duck then you are willing to waltz C If you are willing to waltz then you aren t a duck D A B amp C are all equivalent to the given statement 3 Select the statement that is NOT equivalent to quotIf I don39t invest wisely then I39ll lose my money quot A I invest wisely or I lose my money B If I don39t lose my money then I invested wisely C I lose my money or I invest wisely D If I invest wisely then I won t lose my money World Wide Web Note For practice problems involving negations equivalencies DeMorgan39s Laws and variations on the conditional statement visit the companion website and try THE IMPLICATOR EXAMPLE 2212 Select the statement that is logically equivalent to quotIf all of my friends got hired then some losers are gainfully employed quot A If some losers are not gainfully employed then none of my friends got hired B If no losers are gainfully employed then some of my friends didn39t get hired C If some of my friends didn t get hired then no losers are gainfully employed D If none of my friends got hired then some losers aren39t gainfully employed EXAMPLE 2213 A passage from Lewis Carroll quotAnd now if e39er by chance I put My fingers into glue 0r madly squeeze a righthand foot Into a lefthand shoe Or if I drop upon my toe A very heavy weight I weepfor it reminds me so Of that old man I used to know quot Select the statement that is equivalent to quotIf I put my fingers into glue or squeeze a right hand foot into a left hand shoe or drop a heavy weight upon my toe then I weep quot A If I don39t weep then I don39t put my fingers into glue or squeeze a right hand foot into a left hand shoe or drop a heavy weight upon my toe B I put my fingers into glue or squeeze a right hand foot into a left hand shoe or drop a heavy weight upon my toe and I don39t weep C I don39t put my fingers into glue and I don39t squeeze a right hand foot into a left hand shoe and I don39t drop a heavy weight upon my toe or I weep D If I weep then I put my fingers into glue or squeeze a right hand foot into a left hand shoe or drop a heavy weight upon my toe EXAMPLE 2214 quotI didn t know I was to have a party at all quot said Alice quotbut if there is to be one I think I ought to invite the guests quot Select the statement that is the negation of quotI didn39t know I would have a party but if I will have a party I will choose the guestsquot A I knew I would have a party and if I won t have a party then I won t choose the guests B I knew I would have a party or I will have a party but I won t choose the guests C I knew I would have a party or if I won t have a party then I won t choose the guests D I knew I would have a party and I will have a party but I won t choose the guests PRACTICE EXERCISES 1 Suppose p is the statement You eat carrots and q is the statement You have good eyesight Select the correct statement corresponding to the symbols p gtq A You don t eat carrots or you have good eyesight B If you don t eat carrots then you have good eyesight C It is not the case that if you eat carrots then you have good eyesight D It is not the case that either you eat carrots or you have good eyesight 2 Suppose p is the statement I pass my math course and q is the statement I will change my major to nuclear physics Select the correct statement corresponding to the symbols NP l A It is not the case that if I pass my math course then I will change my major to nuclear physics B It is not the case that either I pass my math course or I will change my major to nuclear physics C I don t pass my math course and I will change my major to nuclear physics D None of these 3 Suppose p is the statement You get a speeding ticket and q is the statement Your insurance rate goes up Select the correct symbolization for the statement If your insurance rate doesn t go up then you don t get a speeding ticket A p gtq B qAp C q gtp D None of these 4 Suppose p is true q is false s is true Find the truth value of SAp gtq gts 5 Suppose p is true q is true s is false Find the truth value of s gtp gtq gts 6 Suppose p is false q is true r is true s is false Find the truth value of pAq gtr gts 7 Suppose p is false q is true s is true Find the truth value of pvq gtqu 8 Suppose p is false q is true r is true s is true Find the truth value of p gtq v rvs 9 Suppose p is false q is true s is false Find the truth value of s gtp v q gts 1018 Make a truth table for the given statement 10 pVq gt pvr 11 pAq gt p gtq 12 pVq gt p 13 pAq gt p gtq 14 pvq gt q 15 pvq gt r 16 pAq gt q 17 pAq gt r 18 pAq gt p 1923 Decide Whether the given statement is true or false 19 True or false pAq gt p gtq is a tautology Hint refer to the answer to 11 above 20 True or false p gtqspAq gt q Hint refer to the answers to 11 and 16 above 21 True or false pAq gt r is a tautology Hint refer to the answer to 1 7 above 22 True or false pvq gt q Epvq gt p Hint refer to the answers to 14 and 12 above 23 True or false pvq gt q is a tautology Hint refer to the answer to 14 above 24 Select the statement that is the negation of If you are a fish then you have cold lips A You are a fish and you don t have cold lips B You are a fish or you don t have cold lips C You aren t a fish or you have cold lips D If you aren t a fish then you don t have cold lips 25 Select the statement that is logically equivalent to Class is cancelled or this is not my lucky day A Class is cancelled and this is not my lucky day B Class is not cancelled and this is my lucky day C Class is not cancelled or this is my lucky day D If class is not cancelled then this is not my lucky day 26 Select the statement that is logically equivalent to If an offer sounds too good to be true then I m interested A An offer sounds too good to be true or I m interested B An offer doesn t sound too good to be true or I m interested C An offer sounds too good to be true and I m not interested D If I m interested in an offer then it sounds too good to be true 27 Select the statement that is the negation of If my computer breaks then I won t be able to waste so much paper A If my computer doesn t break then I will be able to waste so much paper B My computer breaks and I am able to waste so much paper C My computer doesn t break or I am able to waste so much paper D If I am able to waste so much paper then my computer didn t break 28 Select the statement that is logically equivalent to If you want to be on my team then you like getting bossed around A If you don t like getting bossed around then you don t want to be on my team B If you don t want to be on my team then you don t like getting bossed around C If you like getting bossed around then you want to be on my team D A B amp C are all correct E Stop whining and get to work 29 Select the statement that is the negation of Some of us don t have our textbooks A None of us have our textbooks B Some of us have our textbooks C All of us have our textbooks D We have this website instead 30 Select the statement that is logically equivalent to If you have passed MAC4411 then you can t receive credit for MGF1106 A You haven t passed MAC4411 or you can t receive credit for MGF1106 B If you can receive credit for MGF1106 then you haven t passed MAC4411 C All of those who have passed MAC4411 are ineligible to receive credit for MGF1106 D A B and C are all correct 31 Select the statement that is logically equivalent to You can pick your friends or you can pick your nose A You can t pick your friends and you can t pick your nose B You can t pick your friends or you can t pick your nose C If you can t pick your friends then you can pick your nose D but you can t pick your friend s nose 32 Select the statement that is logically equivalent to If you eat that day old burrito you will use lots of hot sauce A If you didn t use lots of hot sauce then you didn t eat that day old burrito B If you don t eat that day old burrito then you won t use lots of hot sauce C If you used lots of hot sauce then you ate that day old burrito D A B amp C are all equivalent to the given statement 33 Select the statement that is the negation of All bulldogs are sweet and some poodles are mean A No bulldogs are sweet and some poodles aren t mean B No bulldogs are sweet or some poodles aren t mean C Some bulldogs aren t sweet and no poodles are mean D Some bulldogs aren t sweet or no poodles are mean 34 Select the statement that is the negation of If some bees fly into your face then all of your plans for the day are ruined A If no bees fly into your face then all of your plans for the day are ruined B If some bees fly into your face then some of your plans for the day aren t ruined C Some bees fly into your face and some of your plans for the day aren t ruined D No bees fly into your face and none of your plans for the day are ruined 35 Select the statement that is logically equivalent to If all of us are OK then all of them are losers A If all of them are losers then all of us are OK B Some of us are OK and all of them are losers C If some of them aren t losers then some of us aren t OK D If some of us aren t OK then some of them aren t losers 36 Select the statement that is logically equivalent to If I lock my cat in the house then she beats up the dog A I lock my cat in the house and she doesn t beat up the dog B I don t lock my cat in the house or she beats up the dog C If I don t lock my cat in the house then she doesn t beat up the dog D None of these 37 Select the statement that is the negation of If all things are considered then I listen to public radio A If I don t listen to public radio then some things aren t considered B If all things are considered then I don t listen to public radio C Some things aren t considered or I listen to public radio D All things are considered and I don t listen to public radio 38 Select the statement that is logically equivalent to We make a first down or we punt A If we don t make a first down then we punt B We punt or we make a first down C Both A amp B D None of these 39 Select the statement that is the negation of No campaign promises are sincere A Some campaign promises are sincere B Some campiagn promises are insincere C All campaign promises are insincere D All camping prom roses are sinister 40 Select the statement that is logically equivalent to quotNo elephants are forgetful quot A If you aren39t an elephant then you are forgetful B If you are an elephant then you aren t forgetful C If you aren39t forgetful then you are an elephant D All of these 41 Referring to 28 select the converse of the given statement 42 Referring to 30 select the contrapositive of the given statement 43 Referring to 37 select the contrapositive of the given statement 44 Referring to 35 select the inverse of the given statement 45 Select the statement that is equivalent to quotNo beggars are choosers quot A If you aren39t a beggar then you are a chooser B All beggars are choosers C If one is a beggar then one isn39t a chooser D One is a beggar and one isn39t a chooser 46 Referring to 32 select the inverse of the given statement ANSWERS TO LINKED EXAMPLES EXAMPLE 223 pvq gtrvs gtw EXAMPLE 224 EXAMPLE 225 1 T 2 T EXAMPLE 226 F T F T F T T F F F T T EXAMPLE 227 EXAMPLE 228 D 2 C 3 B EXAMPLE 229 C 2 B 3 A EXAMPLE 2210 C 2 B 3 D C 2 C 3 D EXAMPLE 2212 EXAMPLE 2213 EXAMPLE 2214 D B 1 1 1 EXAMPLE 2211 1 B C ANSWERS TO PRACTICE EXERCISES 1 C 2 D 4 Suppose p is true q is false s is true Then SAp gtq gts is T 5 Suppose p is true q is true s is false Then s gtp gtq gts is F 6 Suppose p is false q is true r is true s is false Then pAq gtr gts is F 7 Suppose p is false q is true s is true Then pvq gtqu is T 8 Suppose p is false q is true r is true s is true Then p gtq v rvs is T 9 Suppose p is false q is true s is false Then s gtp v q gts is F m pvq a pvr p q r p q r pvq vr Ayxgt Fvq 1 F F 1 F TTTF 1 F TFTF FFFTTT 11 waepav 2 q p q pm 1Igtq Ml 4441 11 1 1 1 F F FTTF lZpw quotP Pvqgty7 p q p q p q lt1wqgt 4 111 F 1 F F FTTF 13 W a Fag 1 p q M1 vaq Hum 44114117 1 1 1 F F FTTF FFTT Npvqaq p q p 11 qu lty qgt qq 11 F F 1 F mm 14m 4 11 gt111 4 11 1 F 1 q PM m P d l lm Im l Ispwgtaql F q Mq wwq Am ltpAqgt 4m 1 1 p p p 1 p 1 F 1 F F 1 gtlgtngtn 17 pAq a x

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