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# NATURE OF MATHEMATCS MATH 1100

LSU

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This 55 page Class Notes was uploaded by Madison Gottlieb Sr. on Tuesday October 13, 2015. The Class Notes belongs to MATH 1100 at Louisiana State University taught by R. Perlis in Fall. Since its upload, it has received 10 views. For similar materials see /class/222623/math-1100-louisiana-state-university in Mathematics (M) at Louisiana State University.

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Date Created: 10/13/15

Negating an Implication We have seen that qu is logically equivalent to pq And we have seen that p q is logically equivalent to p q Question How do you negate gt q You might think that the negation of an implication is another implication but it isn t In fact since p gtq is logically equivalent to the OR statement p V q the negation pgtq is logically equivalent to the negation p V q This is the same as p The dOUble negation just 7 0gtCl is logically equivalent to p We havejust learned that the negation of an implication is an AND statement Example Negate the statement If it will rain I will go to the movies Here p It will rain and q lwill go to the movies The negation of pgtq is p q which is It will rain and I will not go to the movies Example Negate the sentence Studying daily implies you will get good grades Let p You will study daily q You will get good grades The given sentence is pgtq lts negation is p q which is the sentence You will study daily and you will not get good grades Converse and Contrapositive In this lesson we learn two ways to create a new implication from a given one Given a statement If p then q the converse is the statement If q then p obtained by switching the positions of p and q Example Let p and q be the statements p l have brown hair 39 q l have brown eyes The implication If p then q is the statement A Ifl have brown hair then I have brown eyes The converse is the statement B If I have brown eyes then I have brown hair We will see that the truth or falsity of A has little connection to the truth of falsity of B Here are the truth tables for A and for B A B p q if p then q if q then p T T T T T F F T F T T F F F T T The last two columns are different so the truth table for A is not the same as that for B It happens very frequently in ordinary speech as well as in mathematics that people confuse an implication with its converse Try to be aware of this common but serious error so you don t make it The contrapositive of the implication A lfpthen q is the statement C If q then p To obtain the contrapositive of A negate both p and q and then reverse their positions Let p q be the statements from before p I have brown hair q I have brown eyes A f have brown hair then I have brown eyes The contrapositive of A is B f don t have brown eyes then I don t have brown hair The contrapositive sounds complicated Its importance comes from the fact that the contrapositive is logically equivalent to the original implication One is true precisely when the other is true Here are the truth tables A B l0 q q p poq qop T T F F T T T F T F F F F T F T T T F F T T T T We see that the last two columns are identical which says that poq and its contrapositive qop have the same truth values In other words they are logically equivalent although they sound quite different Observe the converse of the converse is the original implication And the contrapositiveof the Con Elfaloositive is the Original i39mpncatign Let s talk about food A survey of 82 students revealed that 0 55 students liked hamburgers 0 31 students liked crawfish 0 9 liked both How many liked hamburgers but not crawfish How many didn t like either choice First he drew a rectangle to represent the students surveyed 39 Hx B l I l The blue circle labelled A denotes the set of students who like hamburger Note that the set A includes the pink region in the middle There are 55 students inside the circle rnarked A The red circle labelled B denotes the set of students who like crawfish Note that the set B includes the pink region in the middle There are 31 students inside the circle rnarked B y A 22 The shaded area is the intersection Ah B It represents the set of students who like hamburgers AND crawfish There are 9 of these The black dot denotes a student who likes hamburgers but NOT crawfish There are 55 9 46 of these There are 31 922 students who like crawfish but not hamburgers This accounts for 4692277 of the 82 students surveyed B La 2 a 22 1 This accounts for 4622977 Of the 82 students surveyed So there are 82 775 students who like neither Choice 5 i l a B y a 22 3 There is something else we can see on this picture The set A has cardinality A 55 and the set B has B 31 So AB 553186 Can you see that the 9 elements in the pink region were counted twice The union Ag B has cardinality Ag B 46922 77 if as B 4 i a 22 1 The union Ag B has cardinality Ag Bl 46922 77 The difference is 86 77 9 These are the 9 pink elements Once again we see that AB lAg BAh Bl This relation between cardinalities is called the principle of inclusion exclusion It holds for any two sets whatsoever A New Pro en f bgw In a class of fteen students there are seven students who play piano siX who play Violin and two who play both Four students play bassoon one plays all three instruments and two play bassoon and piano Two students play Violin but not piano nor bassoon How many students play Violin and bassoon but not piano How many students play none of the three instruments We can code this information as fo ows U15 P B2 P 7 Vi01inistsnotinPnorinB 2 Wiz B 4 mnvwzz P V B1 w W 7 k 0 There is a lot of information here 0 Let s make a Venn diagram 0 Start with a rectangle representing all students in this Class 0 We need 3 circles one for pianists one fOr Violinists one for bassoonists Of the 450 students in a Class 0 There are 302 Freshman 0 There are 37 athletes 0 There are 220 Freshman women 0 There are 16 women athletes 0 There are 24 Freshman athletes 0 Of the athletes 32 are Freshmen or women 0 There are 50 non athlete upper Class men How many women are taking the Class 0 There is a lot of information herei 0 Let s make a Venn diagram 0 Start with a rectangle representing all students in this Class 0 We need 3 circles one for Freshmen one for women one for athletes MA 1100 Students Freshmen I I Athletes Fr Wm A ll Fr Wm Wm Athl Created using the onffne CreateAVenn system at wwwyemndfagramoom REMEMBER start from the CENTER The central region represents students who are Freshmen women and athletes Call this unknown number X 0 Let s add SOME of the numerical information MA 1100 Students Freshmen Athletes Creaied using the online c A v sysrem cquot quot mm 0 There are 16 women athletes Of these X are Freshmen so 16 X are not Freshmen MA 1100 Students Freshmen Athletes Creaied using me online c 4 A v ywm u quot mm 0 There are 24 Freshmen athletes Of these 24 X are not women 0 Of the athletes 32 are either women or Freshmen so 3224 XX16 X Thus 3240 X so X8 There are 220 Freshmen women Of these 8 are athletes so 2208 212 are not athletes MA 1100 Students Freshmen Athletes Creaied usirg ihe saline quot quot sysl em cquot There are 50 male upperclassmen who are non athletes MA 1 100 Students There are 302 212816 Freshmen V Athletes 30223666 male Freshmen who are not athletes Created using the online C quot system an 439 1885 are male and not Freshmen This accounts for 365 of the 450 students The missing students are the upperclass non athlete women There are 45o 36585 of them So there are 2128885 313 women all together MA 1100 Students Alhlehs Created using the online C a quot system a John Venn39s mother Martha S kes came from Swanlandf England and died While he was still uite a young boy His ather was the Rev Henry Venn John39s grandfather was the Rev John Venn who successfully campaigned for the abolition of slavery advocated prison reform and the prevention of cruel SpOI tS Died 4 April 1923 in Cambridge England Io hn Venn Born 4 Aug 1834 in Hull England 0 Venn is best known to mathematicians and logicians for his way of representing sets and their unions and intersections He considered three discs R S and T as t ical subsets of a set U The intersections of these iscs and their complements divide U into 8 non overlapping regions the unions of which give 256 different combinations of the original sets R S T Adapted from O39Connor and E F Robertson October 2003 in http www historymcsst andrewsacuk Mathematicians Vennhtm1 Bertrand Russell 18721970 Bertrand Russell was a aming English liberal who advocated free sex among consenting partners and was therefore banned from entering the US for many years because of his immoral in uence on young women He was also a brilliant Iogician Once when explaining that a false premise implies any conclusion whatsoever he was challenged to explain how the premise i2 could possibly lead to the conclusion Bertrand Russell God i Russell who was very quickwittedt answered immediately God and I are two Therefore God and I are one Admittedly this true story is a bit of a joke sinoe a sentence involving the word God is probably ambiguous do any two people agree on every aspect of the word but it points out why it is valid to say that a false premise implies any conclusion r L Sir Winston Churchill 1874 51 Assuming that 0 1 I prove that Sir Winston Churchill is a carrot Proof First let us agree that a oanot is any object with a green leafy top and an orange body that tapers to a point Remember that we are allowed to use 01 Sinoe the number of leafy tops on Sir Winston is 0 his number of leafy tops is 1 Sir Winston s waist was a rather large 55 inches Multiplying each side of 10 by 55 shows that 55O so his body tapers to a point And Sir Winston was light pink in color which means that the light re ected from his body was had a particular wavelength Let w denote the wavelength of the light in millimeters re ected from Sir Winston Multiplying i0 by w shows that w0 Let d denote the wavelength of light reflected by a carrot Multiplying i0 by d shows that d0 So wd which means that Sir Winston is the same color as the carrot which is orange So Sir Winston has a green leafy top has a body that tapers to a point and is orange Sir Winston Churchill is a carrotw Tautologies Contradictions and If and only ifquot gt A taulology is a usually compound logical sentence which is always true independent of the truth or falsity of the component sentences gt The basic example of a tautology is 39pVp39 If p is T then p is F so quotpVpquot is T And if p is F then p is T so pvp j 1 It is worthwhile making a formal truth table for pVp Here it is 3 1 WT T F T F T T While the components p p can be T or F the sentence pVp is always T So pVp is a tautology KM Technically a single true sentence such as 3 gt 1 is also a tautology But we would usually just call this a true statement and reserve the word tautology39 for compound statements statements built out of two or more other statements which are always true gt A contradiction is a sentence that is always false independent of the truth or falsity of its component sentences 3 The basic example of a contradiction is up A ap If p is T then p is F so the and statement is F if p is F then p is T so the and statement is F The truth table for pp is I0 p pp T F F F T F Since pp is always F we say that pp is a contradiction We have already learned the implication D gt Cl read p implies q We now learn the symbol a which is a double implication Wewritep b q read p ifand only ifqquotto mean p implies q A q implies p quot Here is the truth table for quotp 4 q P 111114 39l39ll l39It cI Pgtq qgtP 9 II39l39II 4114 411114 Observe that the statement pen is T when p and q have the same truth value both T or both F Negations Begin with an OR statement qu The negation of this statement is qu which is read It is not the case thatqu quot Now ququot is T when at least one of p q is true So its negation qu is false when at least one of p q is true Thus jpyq is true when both p q are false e We are seylng that qu has the same truth value as plq Here ls the truth table P q P q PVC qu 3 A TTFFT F F TFFTT F F FTTFT F F FFTTF 39r T The last two column are the same so the negation of We ls logically equivalent to p A q To negate qu negate both of p q and replace V with A rm gt Now look at the AND statement pAq It39s negation is pAq This is the statement It is not the case that both p q are true This new statement is true when atleastoneofp q lsfalse Here lsthetruthtable P Q P 0 PM 9N0 P V q T T F F T F F T F F T F T T F T T F F T T F F T T F T T We see that the last two column are the same so pAq is logically equivalent q

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