Solution Manual for A Transition to Advanced Mathematics, 7th Edition

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CONTENTSChapter 1.1 Propositions and Connectives11.2 Conditionals and Biconditionals61.3 Quantifiers131.4 Basic Proof Methods I171.5 Basic Proof Methods II221.6 Proofs Involving Quantifiers261.7 Additional Examples of Proofs29Chapter 2.1 Basic Concepts of Set Theory382.2 Set Operations402.3 Extended Set Operations and Indexed Families of Sets462.4 Mathematical Induction492.5 Equivalent Forms of Induction592.6 Principles of Counting62Chapter 3.1 Cartesian Products and Relations673.2 Equivalence Relations703.3 Partitions753.4 Ordering Relations783.5 Graphs82Chapter 4.1 Functions as Relations854.2 Constructions of Functions884.3 Functions That Are Onto; One-to-One Functions924.4 One-to-One Correspondences and Inverse Functions954.5 Images of Sets1994.6 Sequences102Chapter 5.1 Equivalent Sets; Finite Sets1055.2 Infinite Sets1085.3 Countable Sets1115.4 The Ordering of Cardinal Numbers1145.5 Comparability of Cardinal Numbers and the Axiom of Choice117Chapter 6.1 Algebraic Structures1196.2 Groups1226.3 Subgroups1266.4 Operation Preserving Maps1286.5 Rings and Fields131Chapter 7.1 Completeness of the Real Numbers1337.2 The Heine-Borel Theorem1367.3 The Bolzano-Weierstrass Theorem1397.4 The Bounded Monotone Sequence Theorem1407.5 Equivalents of Completeness143

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1Logic and Proofs1.1Propositions and Connectives1.(a) true(b) false(c) true(d) false(e) false(f) false(g) false(h) false2.(a) Not a proposition(b) False proposition(c) Not a proposition. It would be a proposition if a value forxhad beenassigned.(d) Not a proposition. It would be a proposition if values forxandyhad beenassigned.(e) False proposition(f) True proposition(g) False proposition(h) True proposition(i) False proposition(j) Not a proposition. It is neither true nor false.3.(a)P∼PP∧ ∼PTFTFTF(b)P∼PP∨ ∼PTFTFTT(c)PQ∼QP∧ ∼QTTFFFTFFTFTTFFTF(d)PQ∼QQ∨ ∼QP∧(Q∨ ∼Q)TTFTTFTFTFTFTTTFFTTF(e)PQ∼QP∧Q(P∧Q)∨ ∼QTTFTTFTFFFTFTFTFFTFT(f)PQP∧Q∼(P∧Q)TTTFFTFTTFFTFFFT1

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1LOGIC AND PROOFS2(g)PQR∼QP∨ ∼Q(P∨ ∼Q)∧RTTTFTTFTTFFFTFTTTTFFTTTTTTFFTFFTFFFFTFFTTFFFFTTF(h)PQ∼P∼Q∼P∧ ∼QTTFFFFTTFFTFFTFFFTTT(i)PQRQ∨RP∧(Q∨R)TTTTTFTTTFTFTTTFFTTFTTFTTFTFTFTFFFFFFFFF(j)PQRP∧QP∧R(P∧Q)∨(P∧R)TTTTTTFTTFFFTFTFTTFFTFFFTTFTFTFTFFFFTFFFFFFFFFFF4.(a) false(b) true(c) true(d) true(e) false(f) false(g) false(h) false(i) true(j) true(k) false(1) false5.(a) No solution.(b)PQP∨QQ∨PTTTTFTTTTFTTFFFFSince the third and fourth columns are the same, the propositions areequivalent.

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1LOGIC AND PROOFS3(c)PQP∧QQ∧PTTTTFTFFTFFFFFFFSince the third and fourth columns are the same, the propositions areequivalent.(d)PQRQ∨RP∨(Q∨R)P∨Q(P∨Q)∨RTTTTTTTFTTTTTTTFTTTTTFFTTTFTTTFTTTTFTFTTTTTFFFTTTFFFFFFFSince the fifth and seventh columns are the same, the propositions areequivalent.(e)PQRQ∧RP∧(Q∧R)P∧Q(P∧Q)∧RTTTTTTTFTTTFFFTFTFFFFFFTFFFFTTFFFTFFTFFFFFTFFFFFFFFFFFFFSince the fifth and seventh columns are the same, the propositions areequivalent.(f)PQRQ∨RP∧(Q∨R)P∧QP∧R(P∧Q)∨(P∧R)TTTTTTTTFTTTFFFFTFTTTFTTFFTTFFFFTTFTTTFTFTFTFFFFTFFFFFFFFFFFFFFFSince the fifth and eighth columns are the same, the propositions areequivalent.(g)PQRQ∧RP∨(Q∧R)P∨QP∨R(P∨Q)∧(P∨R)TTTTTTTTFTTTTTTTTFTFTTTTFFTFFFTFTTFFTTTTFTFFFTFFTFFFTTTTFFFFFFFF

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1LOGIC AND PROOFS4Since the fifth and eighth columns are the same, the propositions areequivalent.(h) No solution.(i)PQP∨Q∼(P∨Q)∼P∼Q∼P∧ ∼QTTTFFFFFTTFTFFTFTFFTFFFFTTTTSince the fourth and eighth columns are the same, the propositions areequivalent.6.(a) equivalent(b) equivalent(c) equivalent(d) equivalent(e) equivalent(f) not equivalent(g) not equivalent(h) not equivalent7.(a)∼P, true(b)P∧Q, true(c)P Q, true(d)P∨Q∨R, true8.(a) SincePis equivalent toQ,Phas the same truth table asQ. Therefore,Qhas the same truth table asP, soQis equivalent toP.(b) SincePis equivalent toQ,PandQhave the same truth table. SinceQisequivalent toR,QandRhave the same truth table. Thus,PandRhavethe same truth table soPis equivalent toR.(c) SincePis equivalent toQ,PandQhave the same truth table. That is, thetruth table forPhas value true on exactly the same lines that the truthtable forQhas value true. Therefore the truth table for∼Qhas value falseon exactly the same lines that the truth table for∼Phas the value false.Thus∼Qand∼Phave the same truth table.9.(a) (P∧Q)∨(∼P∧ ∼Q) is neither.PQ∼P∼QP∧Q∼P∧ ∼Q(P∧Q)∨(∼P∧ ∼Q)TTFFTFTFTTFFFFTFFTFFFFFTTFTT(b)∼(P∧ ∼P) is a tautology.P∼PP∧ ∼P∼(P∧ ∼P)TFFTFTFT(c) (P∧Q)∨(∼P∨ ∼Q) is a tautology.PQ∼P∼QP∧Q∼P∨ ∼Q(P∧Q)∨(∼P∨ ∼Q)TTFFTFTFTTFFTTTFFTFTTFFTTFTT(d) (A∧B)∨(A∧ ∼B)∨(∼A∧B)∨(∼A∧ ∼B) is a tautology.

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1LOGIC AND PROOFS5(A∧B)∨(A∧ ∼B)∨AB∼A∼BA∧BA∧ ∼B∼A∧B∼A∧ ∼B(∼A∧B)∨(∼A∧ ∼B)TTFFTFFFTFTTFFFTFTTFFTFTFFTFFTTFFFTT(e) (Q∧ ∼P)∧ ∼(P∧R) is neither.PQR∼PQ∧ ∼PP∧R∼(P∧R)(Q∧ ∼P)∧ ∼(P∧R)TTTFFTFFFTTTTFTTTFTFFTFFFFTTFFTFTTFFFFTFFTFTTFTTTFFFFFTFFFFTFFTF(f)P∨[(∼Q∧P)∧(R∨Q)] is neither.PQR∼Q∼Q∧PR∨Q[(∼Q∧P)∧(R∨Q)]P∨[(∼Q∧P)∧(R∨Q)]TTTFFTFTFTTFFTFFTFTTTTTTFFTTFTFFTTFFFTFTFTFFFTFFTFFTTFFTFFFTFFFF10.(a) contradiction(b) tautology(c) tautology(d) tautology11.(a)xis not a positive integer.(b) Cleveland will lose the first game and the second game. Or, Cleveland willlose both games.(c) 5<3(d) 641,371 is not composite. Or 641,371 is prime.(e) Roses are not red or violets are not blue.(f)Tis bounded andTis not compact.(g)Mis not odd orMis not one-to-one.(h) The function.fdoes not have a positive first derivative atxor does nothave a positive second derivative atx.(i) The functiongdoes not have a relative maximum atx= 2 (deleted comma)and does not have a relative maximum atx= 4, or elsegdoes not have arelative minimum atx= 3.(j)z < sorz≤t.(k)Ris not transitive orRis reflexive.(l) If the functionghas a relative minimum atx= 2 orx= 4, thengdoesnot have a relative minimum atx= 3.12.(a) [∼(∼P)]∨[(∼Q)∧(∼S)](b) [Q∧(∼S)]∨ ∼(P∧[Q∧(∼S)]∨ ∼((∼P∧Q).(c) [[P∧(∼Q)]∨[(∼P)∧(∼R)]]∨[(∼P)∧S](d) [(∼P)∨([Q∧(∼(∼P))]∧Q)]∨R.

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1LOGIC AND PROOFS613.(a)i.ABA©∨BTTFFTTTFTFFFii.ABA∨BA∧B∼(A∧B)(A∨B)∧ ∼(A∧B)TTTTFFFTTFTTTFTFTTFFFFTFSince the final columns of the two tables are identical, the two propo-sitions have the same truth table, thus they are equivalent.(b)i.ABANANDBANORBTTFFFTTFTFTFFFTTii.ABANANDBANORB(ANANDB)∨(ANORB)TTFFFFTTFTTFTFTFFTTTSince the third and last columns are equal, the propositions are equiv-alent.iii.ABANANDBANORB(ANANDB)∧(ANORB)TTFFFFTTFFTFTFFFFTTTSince the fourth and last columns are equal, the propositions areequivalent.1.2Conditionals and Biconditionals1.(a) Antecedent: squares have three.Consequent: triangles have four sides.(b) Antecedent: The moon is made of cheese.Consequent: 8 is an irrational number.(c) Antecedent:bdivides 3.Consequent:bdivides 9.(d) Antecedent:fis differentiable.Consequent:fis continuous.(e) Antecedent:ais convergent.Consequent:ais bounded.(f) Antecedent:fif integrable.Consequent:fis bounded.(g) Antecedent: 1 + 1 = 2.Consequent: 1 + 2 = 3.

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1LOGIC AND PROOFS7(h) Antecedent: the fish bite.Consequent: the moon is full.(i) Antecedent: An athlete qualifies for the Olympic team.Consequent: The athlete has a time of 3 minutes, 48 seconds or less ( inthe event).2.(a) Converse: If triangles have four sides, then squares have three sides.Contrapositive: If triangles do not have four sides, then squares do not havethree sides.(b) Converse: If 8 is irrational, then the moon is made of cheese.Contrapositive: If 8 is rational, then the moon is not made of cheese.(c) Converse: Ifbdivides 9, thenbdivides 3.Contrapositive: Ifbdoes not divide 9, thenbdoes not divide 3.(d) Converse: Iffis continuous, thenfis differentiable.Contrapositive: Iffis not continuous, thenfis not differentiable.(e) Converse: Ifais bounded, thenais convergent.Contrapositive: Ifais not bounded, thenais not convergent.(f) Converse: Iffis bounded, thenfis integrable.Contrapositive: Iffis not bounded, thenfis not integrable.(g) Converse: If 1 + 2 = 3, then 1 + 1 = 2.Contrapositive: If 1 + 1= 2, then 1 + 2= 3.(h) Converse: If the moon is full, then fish will bite.Contrapositive: If the moon is not full, then fish will not bite.(i) Converse: A time of 3 minutes, 48 seconds or less is sufficient to qualify forthe Olympic team.Contrapositive: If an athlete records a time that is not 3 minutes and 48seconds or less, then that athlete does not qualify for the Olympic team.3.(a)Qmay be either true or false.(b)Qmust be true.(c)Qmust be false.(d)Qmust be false.(e)Qmust be false.4.(a) Antecedent:A(x) is an open sentence with variablex.Consequent:∼(∀x)A(x) is equivalent to (∃x)∼A(x).(b) Antecedent: Every even natural number greater than 2 is the sum of twoprimes.Consequent: Every odd natural number greater than 5 is the sum of threeprimes.(c) Antecedent:Ais a set withnelements.Consequent:P(A) is a set with 2nelements.(d) Antecedent:Sis a subset ofNsuch that 1∈Sand, for alln∈N, ifn∈S,thenn+ 1∈S.Consequent:S=N.(e) Antecedent:Ais a finite set withmelements andBis a finite set withnelements.Consequent:A×B=mn.

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1LOGIC AND PROOFS8(f) Antecedent:Ris a partial order forAandB⊆A.Consequent: If sup(B) exists, it is unique.(g) Antecedent:A,B,C, andDare sets,fis a function fromAtoB,gis afunction fromBtoC, andhis a function fromCtoD.Consequent: (h◦g)◦f=h◦(g◦f).(h) Antecedent:AandBare disjoint finite sets.Consequent:A∪Bis finite andA∪B=A+B.5.(a) true(b) false(c) true(d) true(e) true(f) true(g) true(h) false6.(a) true(b) true(c) true(d) true(e) false(f) true(g) false(Thesymbolforhelium is He.)(h) true(i) false(j) true(k) false7.(a)PQQ∧PP⇒(Q∧P)TTTTFTFTTFFFFFFT(b)PQ∼P∼P⇒QQ⇔P(∼P⇒Q)∨(Q⇔P)TTFTTTFTTTFTTFFTFTFFTFTT(c)PQ∼QQ⇔P∼Q⇒(Q⇔P)TTFTTFTFFTTFTFFFFTTT(d)PQP∨QP∧Q(P∨Q)⇒(P∧Q)TTTTTFTTFFTFTFFFFFFT(e)PQRP∧QQ∧RP∨R(P∧Q)∨(Q∧R)(P∧Q)∨(Q∧R)⇒P∨RTTTTTTTTFTTFTTTTTFTFFTFTFFTFFTFTTTFTFTTTFTFFFFFTTFFFFTFTFFFFFFFT

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1LOGIC AND PROOFS9(f)PQRSQ⇒SQ⇒RP∨QS∨R(Q⇒S)∧(Q⇒R)TTTTTTTTTFTTTTTTTTTFTTTTTTTFFTTTTFTTTTFTTFTTFFTFTTFTTFTFFTTTTTTFFFTTTFTTTTTFFTTTFFTTFFTTTFTFTFTTTTTFFTFTTFTTTTFFFFTFFFTFFFFTFFTFFFTTTFTFFFFTTFFTPQRS(P∨Q)⇒(S∨R)[(Q⇒S)∧(Q⇒R)]⇒[(P∨Q)⇒(S∨R)]TTTTTTFTTTTTTFTTTTFFTTTTTTFTTTFTFTTTTFFTTTFFFTTTTTTFTTFTTFTTTFTFTTFFTFTTTTFFFTFTFFFTTFFFFFFFFFTT8.(a)PQ∼PP⇒Q(∼P)∨QTTFTTFTTTTTFFFFFFTTTSince the fourth and fifth columns are the same, the propositionsP⇒Qand (∼P)∨Qare equivalent.(b)PQP⇒QQ⇒PP⇔Q(P⇒Q)∧(Q⇒P)TTTTTTFTTFFFTFFTFFFFTTTTSince the fifth and sixth columns are the same, the propositionsP⇔Qand (P⇒Q)∧(Q⇒P) are equivalent.(c)PQ∼QP⇒Q∼(P⇒Q)P∧ ∼QTTFTFFFTFTFFTFTFTTFFTTFFSince the fifth and sixth columns are the same, the propositions∼(P⇒Q)andP∧ ∼Qare equivalent.

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1LOGIC AND PROOFS10(d)PQ∼P∼QP∧Q∼(P∧Q)P⇒∼QP⇒∼QTTFFTFFFFTTFFTTTTFFTFTTTFFTTFTTTSince the sixth, seventh and eighth columns are the same, all three propo-sitions are equivalent.(e)PQRQ⇒RP⇒(Q⇒R)P∧Q(P∧Q)⇒RTTTTTTTFTTTTFTTFTTTFTFFTTTFTTTFFFTFFTFFTFTTFFTTFTFFFTTFTSince the fifth and seventh columns are the same, the propositions areequivalent.(f)PQRQ∧RP⇒(Q∧R)P⇒QP⇒R(P⇒Q)∧(P⇒R)TTTTTTTTFTTTTTTTTFTFFFTFFFTFTTTTTTFFFTFFFTFFTTTTTFFFFFFFFFFFTTTTSince the fifth and eighth columns are the same, the propositions areequivalent.(g)PQRP∨QP∨Q)⇒RP⇒RQ⇒R(P⇒R)∧(Q⇒R)TTTTTTTTFTTTTTTTTFTTTTTTFFTFTTTTTTFTFFFFFTFTFTFFTFFTFFTFFFFFTTTTSince the fifth and seventh columns are the same, the propositions areequivalent.9. (a) yes(b) no(c) yes(d) yes(e) no(f) no10.(a) [(fhas a relative minimum atx0)∧(fis differentiable atx0)]⇒(f′(x0) =0)(b) (nis prime)⇒[(n= 2)∨(nis odd)](c) (xis irrational)⇒[(xis real)∧ ∼(xis rational)](d) [(x= 1)∨(x=−1)]⇒(|x|= 1)(e) (x0is a critical point forf)⇔[(f′(x0) = 0)∨(f′(x0) does not exist)](f) (Sis compact)⇔[(Sis closed)∧(Sis bounded)](g) (B is invertible)⇔(det B=0)

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1LOGIC AND PROOFS11(h) (6≥n−3)⇒(n >4)∨(n >10)(i) (xis Cauchy)⇒(xis convergent)(j) (limx→x0f(x) =f(x0))⇒(fis continuous atx0)(k) [(fis differentiable atx0)∧(fis strictly increasing atx0)]⇒(f′(x0))11.(a) LetSbe “I go to the store” andRbe “It rains.” The preferred translation:is∼S⇒R(or, equivalently,∼R⇒S). This could be read as “If itdoesn’t rain, then I go to the store.”The speaker might mean “I go to the store if and only if it doesn’t rain(S⇒∼R) or possibly “If it rains, then I don’t go to the store” (R⇒∼S).(b) There are three nonequivalent ways to translate the sentence, using thesymbolsD: “The Dolphins make the playoffs” andB: “The Bears win allthe rest of their games.” The first translation is preferred, but the speakermay have intended any of the three.∼B⇒∼Dor, equivalently,D⇒B∼D⇒∼Bor, equivalently,B⇒D∼B⇔∼Dor, equivalently,B⇔D(c) LetGbe “You can go to the game” andHbe “You do your homeworkfirst.”It is most likely that a student and parent both interpret this statement asa biconditional,G⇔H.(d) LetWbe “You win the lottery” andTbe “You buy a ticket.” Of the threecommon interpretations for the word “unless,” only the form∼T⇒∼W(or, equivalently,W⇒T) makes sense here.12.(a)PQRP∨Q(P∨Q)⇒R∼P∧ ∼Q∼R⇒(∼P∧ ∼Q)TTTTTFTFTTTTFTTFTTTFTFFTFTTTTTFTFFFFTFTFFFTFFTFFFFFFFTTTSince the fifth and seventh columns are the same, (P∨Q)⇒Rand∼R⇒(∼P∧ ∼Q) are equivalent.(b)PQRP∧Q(P∧Q)⇒R∼Q∼RP∧ ∼R(P∧ ∼R)⇒∼QTTTTTFFFTFTTFTFFFTTFTFTTFFTFFTFTTFFTTTFTFFTTFFTFFTFTFTTFFFTTTTTFFFFTTTFTSince the fifth and ninth columns are the same, the propositions (P∧Q)⇒Rand (P∧ ∼R)⇒∼Qare equivalent.

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1LOGIC AND PROOFS12(c)PQRQ∧RP⇒(Q∧R)∼Q∨ ∼R(∼Q∨ ∼R)⇒∼PTTTTTFTFTTTTFTTFTFFTFFFTFTTTTTFFFTFFTFFTTTTFFFFTFFFFFTTTSince the fifth and seventh columns are the same, the propositionsP⇒(Q∧R) and (∼Q∨ ∼R)⇒∼Pare equivalent.(d)PQRQ∨RP⇒(Q∨R)P∧ ∼R(P∧ ∼R)⇒QTTTTTFTFTTTTFTTFTTTFTFFTTTFTTTFTTTTFTFTTFTTFFFFTFFFFFTFTSince the fifth and seventh columns are the same, the propositionsP⇒(Q∨R) and (P∧ ∼R)⇒Qare equivalent.(e)PQRP⇒Q(P⇒Q)⇒RP∧ ∼Q(P∧ ∼Q)∨RTTTTTFTFTTTTFTTFTFTTTFFTTTFTTTFTFFFFTFTFFFTFFFTTTFFFTFFFSince the fifth and seventh columns are the same, the propositions (P⇒Q)⇒Rand (P∧ ∼Q)∨Rare equivalent.(f)PQP⇔Q∼P∨Q∼Q∨P(∼P∨Q)∧(∼Q∨P)TTTTTTFTFTFFTFFFTFFFTTTTSince the third and sixth columns are the same, the propositionsP⇔Qand (∼P∨Q)∧(∼Q∨P) are equivalent.13.(a) If 6 is an even integer, then 7 is an odd integer.(b) If 6 is an odd integer, then 7 is an odd integer.(c) This is not possible.(d) If 6 is an even integer, then 7 is an even integer. (Any true conditionalstatement will work here.)14.(a) If 7 is an odd integer, then 6 is an odd integer.(b) This is not possible.

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1LOGIC AND PROOFS13(c) This is not possible.(d) If 7 is an odd integer, then 6 is an odd integer. (Any false conditionalstatement will work here.)15.(a) Converse: Iff′(x0) = 0, thenfhas a relative minimum atx0and isdifferentiable atx0. False:f(x) =x3has first derivative 0 but no minimumatx0= 0.Contrapositive: Iff′(x0)= 0, thenfeither has no relative minimum atx0or is not differentiable atx0. True.(b) Converse: Ifn= 2 ornis odd, thennis prime. False: 9 is odd but notprime.Contrapositive: Ifnis even and not equal to 2, thennis not prime. True.(c) Converse: Ifxis irrational, thenxis real and not rational. TrueContrapositive: Ifxis not irrational, thenxis not real orxis rational. True(d) Converse: If|x|= 1, thenx= 1 orx=−1. True.Contrapositive: If|x| = 1, thenx= 1 andx=−1. True.16.(a) tautology(b) tautology(c) contradiction(d) neither(e) tautology(f) neither(g) contradiction(h) tautology(i) contradiction(j) neither(k) tautology(l) neither17.(a)PQP⇒Q∼P∼Q∼P⇒∼QTTTFFTFTTTFFTFFFTTFFTTTTComparison of the third and sixth columns of the truth table shows thatP⇒Qand∼P⇒∼Qare not equivalent.(b) We see from the truth table in part (a) that both propositionsP⇒Qand∼P⇒∼Qare true only whenPandQhave the same truth value.(c) The converse ofP⇒QisQ⇒P. The contrapositive of the inverse ofP⇒Qis∼∼Q⇒∼∼P, so the converse and the contrapositive of theinverse are equivalent.The inverse of the contrapositive ofP⇒Qis also∼∼Q⇒∼∼P, so ittoo is equivalent to the converse.1.3Quantifiers1.(a)∼(∀x)(xis precious⇒xis beautiful) or (∃x)(xis precious andxis notbeautiful)(b) (∀x)(xis precious⇒xis not beautiful)(c) (∃x)(xis isosceles∧xis a right triangle)(d) (∀x)(xis a right triangle⇒xis not isosceles) or∼(∃x)(xis a righttriangle∧xis isosceles)(e) (∀x)(xis honest)∨ ∼(∃x)(xis honest)(f) (∃x)(xis honest)∧(∃x)(xis not honest)(g) (∀x)(x= 0⇒(x >0∨x <0))(h) (∀x)(xis an integer⇒(x >−4∨x <6)) or (∀x∈Z)(x >−4∨x <6)

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1LOGIC AND PROOFS14(i) (∀x)(∃y)(x > y)(j) (∀x)(∃y)(x < y)(k) (∀x)(∀y)[(xis an integer∧yis an integer∧y > x)⇒(∃z)(y > z > x)] or(∀x∈Z)(∀y∈Z)[y > x⇒(∃z)(y > z > x)](l) (∃x)(xis a positive integer andxis smaller than all other positive integers)or (∃x)(xis a positive integer and (∀y)(yis a positive integer⇒x≤y))or (∃x∈Z)[x >0∧(∀y∈Z)(y >0⇒y > x)](m) (∀x)(∼(∀y)(xlovesy)) or∼(∃x)(∀y)(xlovesy) or (∀x)(∃y)(xdoes not lovey)(n) (∀x)(∃y)(xlovesy)(o) (∀x)(x >0⇒(∃!y)(2y=x)2.(a) (∀x)(xis precious⇒xis beautiful)All precious stones are beautiful.(b) (∃x)(xis precious∧xis beautiful)There is a beautiful precious stone, or Some precious stones are beautiful.(c)∼(∃x)(xis isosceles andxis a right triangle) or (∀x)(xis not isosceles orxis not a right triangle) or (∀x)(xis right triangle⇒xis not isosceles) or(∀x)(xis isosceles⇒xis not a right triangle).There is no isosceles right triangle.(d) (∃x)(xis isosceles∧xis a right triangle)There is an isosceles right triangle.(e) (∃x)(xis dishonest)∧(∃x)(xis dishonest)Some people are honest and some people are dishonest.(f) (∀x)(xis honest)∨(∀x)(xis dishonest)All people are honest or no one is honest.(g) (∃x)(x= 0∧xis not positive∧xis not negative)There is a nonzero real number that is neither positive nor negative.(h) (∃x)(xis an integer∧x≤ −4∧x≥6)) or (∃x∈Z)(x≤ −4∧x≥6)There is an integer that is less than or equal to−4 and greater than orequal to 6.(i) (∃x)(∀y)(x≤y)Some integer is less than or equal to every integer, or There is a smallestinteger.(j) (∃x)(∀y)(x≥y)Some integer is greater than every other integer, or There is a largestinteger.(k) (∃x)(∃y)[xis an integer∧yis an integery > x∧(∀z)(z≤y∨x≤z)] or(∃x∈Z)(∃y∈Z)[y > x∧(∀z)(z≤y∨x≤z)]There is an integerxand a larger integerysuch that there is no real numberbetween them.(l) (∀x)(xis a positive integer⇒(∃y)(yis a positive integer)∧x > y) or(∀x∈Z)[x≤0∨(∃y∈Z)(y >0∧x > y)]. For every positive integer thereis a smaller positive integer.Or,∼(∃x)(xis a positive integer∧(∀y)(yis a positive integer⇒x≤y)) or∼(∃x∈Z)[x >0∧(∀y∈Z(y >0⇒y > x)]There is no smallest positive integer.

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Algebra II - Rational Expressions

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