Lecture 2: What is Proof?

Transcription

1 Lecture 2: What is Proof? Math /26/16 Webster Proof and Its History 8/ / 1

2 Evolution of Proof Proof, a relatively new idea Modern mathematics could not be supported at its foundation, nor construct the top floor without precise statements Intuition works up to a point Webster Proof and Its History 8/ / 1

3 Evolution of Proof Proof, a relatively new idea Modern mathematics could not be supported at its foundation, nor construct the top floor without precise statements Intuition works up to a point Four Stages : Webster Proof and Its History 8/ / 1

4 Evolution of Proof Proof, a relatively new idea Modern mathematics could not be supported at its foundation, nor construct the top floor without precise statements Intuition works up to a point Four Stages : just answers volumes, areas, lengths, angles, relationships; practical demonstration (intuition) that it works, calculation, formulae verification/justification plausible picture, description, analogy, invoke the gods modern proof Webster Proof and Its History 8/ / 1

5 Egyptians: Practical Questions Right pyramidal frustum Webster Proof and Its History 8/ / 1

6 Egyptians: Practical Questions Right pyramidal frustum Known in 1850 BCE. Requires calculus for proof of derivation V = h 3 (a2 + ab + b 2 ) Webster Proof and Its History 8/ / 1

7 Egyptians: Practical Questions Right pyramidal frustum V = h 3 (a2 + ab + b 2 ) Known in 1850 BCE. Requires calculus for proof of derivation Answer : No record of derivation or proof. Webster Proof and Its History 8/ / 1

8 Euclid: The Outlier Greeks: Geometry, trigonometry, forms, idealizations Webster Proof and Its History 8/ / 1

9 Euclid: The Outlier Greeks: Geometry, trigonometry, forms, idealizations Triangles, rectangles, circles (building/construction) Webster Proof and Its History 8/ / 1

10 Euclid: The Outlier Greeks: Geometry, trigonometry, forms, idealizations Triangles, rectangles, circles (building/construction) The father of rigor (300 BCE): Euclid Precursors: Thales (600 BCE), Pythagoras and Pythagoreans (550 BCE), Eudoxus (400 BCE) calculations and declarations, theorems Webster Proof and Its History 8/ / 1

11 Euclid: The Outlier Greeks: Geometry, trigonometry, forms, idealizations Triangles, rectangles, circles (building/construction) The father of rigor (300 BCE): Euclid Precursors: Thales (600 BCE), Pythagoras and Pythagoreans (550 BCE), Eudoxus (400 BCE) calculations and declarations, theorems Who was Euclid? Webster Proof and Its History 8/ / 1

12 Euclid: The Outlier Greeks: Geometry, trigonometry, forms, idealizations Triangles, rectangles, circles (building/construction) The father of rigor (300 BCE): Euclid Precursors: Thales (600 BCE), Pythagoras and Pythagoreans (550 BCE), Eudoxus (400 BCE) calculations and declarations, theorems Who was Euclid? geometer, archivist, organizer Webster Proof and Its History 8/ / 1

13 Euclid: The Outlier Greeks: Geometry, trigonometry, forms, idealizations Triangles, rectangles, circles (building/construction) The father of rigor (300 BCE): Euclid Precursors: Thales (600 BCE), Pythagoras and Pythagoreans (550 BCE), Eudoxus (400 BCE) calculations and declarations, theorems Who was Euclid? The Elements planar geometry geometer, archivist, organizer Axioms, definitions, theorem, proof Webster Proof and Its History 8/ / 1

14 Euclid: The Outlier Greeks: Geometry, trigonometry, forms, idealizations Triangles, rectangles, circles (building/construction) The father of rigor (300 BCE): Euclid Precursors: Thales (600 BCE), Pythagoras and Pythagoreans (550 BCE), Eudoxus (400 BCE) calculations and declarations, theorems Who was Euclid? The Elements planar geometry geometer, archivist, organizer Axioms, definitions, theorem, proof Demonstration and Justification But where did the proofs come from? Webster Proof and Its History 8/ / 1

15 Dark Ages Webster Proof and Its History 8/ / 1

16 Renaissance Number Theorists Gerolamo Cardano ( ) and Niccolo Tartaglia ( ) Era of the master Webster Proof and Its History 8/ / 1

17 Renaissance Number Theorists Gerolamo Cardano ( ) and Niccolo Tartaglia ( ) Era of the master Solving the cubic equation Forms of solution, authorship, conflict Webster Proof and Its History 8/ / 1

18 Renaissance Number Theorists Gerolamo Cardano ( ) and Niccolo Tartaglia ( ) Era of the master Solving the cubic equation Forms of solution, authorship, conflict Public challenge matches (Tartaglia v. Ferrari) Webster Proof and Its History 8/ / 1

19 Renaissance Number Theorists Gerolamo Cardano ( ) and Niccolo Tartaglia ( ) Era of the master Solving the cubic equation Forms of solution, authorship, conflict Public challenge matches (Tartaglia v. Ferrari) Demonstration and Verification : I have the answer, I can check these cases. Webster Proof and Its History 8/ / 1

20 Calculus Era I Johannes Kepler ( ): Proof, statistical methods Isaac Barrow ( ): Tangent problem, discovered the fundamental theorem of calculus, derive solutions to physical problems Isaac Newton ( ): Student of Barrow s, sloppily axiomatize some ideas in calculus to derive equations and solutions to difficult problems; motivated by infinitesimal change in physical problems Gottfried Leibniz ( ): Philosopher and polymath, looser notation and better intuition provided the backbone of modern analysis Webster Proof and Its History 8/ / 1

21 Calculus Era II Ideas Webster Proof and Its History 8/ / 1

22 Calculus Era II Ideas infinitely small infinitely big infinitesimals, limits Webster Proof and Its History 8/ / 1

23 Calculus Era II Ideas infinitely small infinitely big infinitesimals, limits Big equations/formulae...on shaky footing; Justification and explication Webster Proof and Its History 8/ / 1

24 Calculus Era II Ideas infinitely small infinitely big infinitesimals, limits Big equations/formulae...on shaky footing; Justification and explication Solutions to big problems (e.g., the area problem, the tangent problem, calculus of variations) Not without critics George Berkeley ( ) Transition: Webster Proof and Its History 8/ / 1

25 Calculus Era II Ideas infinitely small infinitely big infinitesimals, limits Big equations/formulae...on shaky footing; Justification and explication Solutions to big problems (e.g., the area problem, the tangent problem, calculus of variations) Not without critics George Berkeley ( ) Transition: Leonhard Euler ( ): Prolific, pioneering...sloppy; intuitive arguments, formal manipulation Webster Proof and Its History 8/ / 1

26 Calculus Era II Ideas infinitely small infinitely big infinitesimals, limits Big equations/formulae...on shaky footing; Justification and explication Solutions to big problems (e.g., the area problem, the tangent problem, calculus of variations) Not without critics George Berkeley ( ) Transition: Leonhard Euler ( ): Prolific, pioneering...sloppy; intuitive arguments, formal manipulation Example: The Basel problem Webster Proof and Its History 8/ / 1

27 Golden Era: 18th and 19th Century Moving towards modern proof Webster Proof and Its History 8/ / 1

28 Golden Era: 18th and 19th Century Moving towards modern proof Joseph Louis Lagrange ( ): Move from blind formal manipulation and intuition, Taylor polynomials Karl Friedrich Gauss ( ): Adequate (modern) analysis of series August Louis Cauchy ( ): French counterpart to Gauss; limits, continuity, differentiation, and definite integral (modern), complex calculus Karl Weierstrass ( ): Topology, foundations, real numbers, real analysis including ε δ Webster Proof and Its History 8/ / 1

29 Modernity I David Hilbert ( ): Last conversant mathematician of entire subject Wanted axiomatic and consistent formulation of set theory Formalism: mathematics is manipulation of symbols according to agreed upon formal rules Hilbert s Program: Formal system which is complete, consistent, decidable Hilbert s 23 Problems at the International Congress in 1900 Axiomatic and consistent construction of the integers What are the rules? and What is the context? Webster Proof and Its History 8/ / 1

30 Modernity II Henri Poincare ( ): Great mathematician; intuition opposite Hilbert and Russell; mathematics fundamentally distinct from logic; proof has intuitive steps, not purely mechanical Nicolas Bourbaki (1930 s): pseudonym for French mathematicians; encyclopedists: consistent, rigorous treatment of all common mathematics; no pictures, no intuition Bertrand Russell ( ): foundations; reducing mathematics to its most elementary grains ; logicism; constructing the integers Kurt Godel ( ): Incompleteness; destruction of Hilbert s Program and Russell/Whitehead Foundationalism/logicism Webster Proof and Its History 8/ / 1

31 Incompleteness In 1931 he proved: For any computable axiomatic system that is powerful enough to describe the the natural numbers/arithmetic (e.g. the Peano axioms or ZFC set theory), that: Webster Proof and Its History 8/ / 1

32 Incompleteness In 1931 he proved: For any computable axiomatic system that is powerful enough to describe the the natural numbers/arithmetic (e.g. the Peano axioms or ZFC set theory), that: If the formal system is consistent, it cannot be complete. Webster Proof and Its History 8/ / 1

33 Incompleteness In 1931 he proved: For any computable axiomatic system that is powerful enough to describe the the natural numbers/arithmetic (e.g. the Peano axioms or ZFC set theory), that: If the formal system is consistent, it cannot be complete. The consistency of the axioms cannot be proven within the system. Webster Proof and Its History 8/ / 1

34 Incompleteness In 1931 he proved: For any computable axiomatic system that is powerful enough to describe the the natural numbers/arithmetic (e.g. the Peano axioms or ZFC set theory), that: If the formal system is consistent, it cannot be complete. The consistency of the axioms cannot be proven within the system. These theorems ended attempts by many of those above to find a small set of consistent axioms sufficient for all of modern mathematics. Webster Proof and Its History 8/ / 1

35 Russell s Two Directions and Weirdness WEIRDNESS Foundations Seemingly Self-Evident Mathematical Facts The Infinite WEIRDNESS Webster Proof and Its History 8/ / 1

36 Russell s Two Directions and Weirdness WEIRDNESS Foundations Seemingly Self-Evident Mathematical Facts The Infinite WEIRDNESS Modern Mathematics: walking on egg shells Webster Proof and Its History 8/ / 1

37 Russell s Two Directions and Weirdness WEIRDNESS Foundations Seemingly Self-Evident Mathematical Facts The Infinite WEIRDNESS Modern Mathematics: walking on egg shells Proof in this class: Pretty axiomatic, basic We will not prove everything, and we will define many things intuitively Take many basic mathematical facts as given May take somethings for granted and come back and prove them later Webster Proof and Its History 8/ / 1

38 Fundamental Terms I Axiom 1 Definition versus Theorem Webster Proof and Its History 8/ / 1

39 Fundamental Terms I Axiom Fundamental building block of a theory, stated in terms outside of the theory; should be consistent with one another 1 Definition versus Theorem Webster Proof and Its History 8/ / 1

40 Fundamental Terms I Axiom Fundamental building block of a theory, stated in terms outside of the theory; should be consistent with one another Definition 1 1 Definition versus Theorem Webster Proof and Its History 8/ / 1

41 Fundamental Terms I Axiom Fundamental building block of a theory, stated in terms outside of the theory; should be consistent with one another Definition 1 Declaration of the properties of a new object, written only in terms of other well-defined objects and axioms 1 Definition versus Theorem Webster Proof and Its History 8/ / 1

42 Fundamental Terms I Axiom Fundamental building block of a theory, stated in terms outside of the theory; should be consistent with one another Definition 1 Declaration of the properties of a new object, written only in terms of other well-defined objects and axioms Theorem 1 Definition versus Theorem Webster Proof and Its History 8/ / 1

43 Fundamental Terms I Axiom Fundamental building block of a theory, stated in terms outside of the theory; should be consistent with one another Definition 1 Declaration of the properties of a new object, written only in terms of other well-defined objects and axioms Theorem A true statement in a theory which relates defined terms and other statements within the theory; 1 Definition versus Theorem Webster Proof and Its History 8/ / 1

44 Fundamental Terms I Axiom Fundamental building block of a theory, stated in terms outside of the theory; should be consistent with one another Definition 1 Declaration of the properties of a new object, written only in terms of other well-defined objects and axioms Theorem A true statement in a theory which relates defined terms and other statements within the theory; Often stated in the form: If p, then q where p and q are statements which are true or false; 1 Definition versus Theorem Webster Proof and Its History 8/ / 1

45 Fundamental Terms I Axiom Fundamental building block of a theory, stated in terms outside of the theory; should be consistent with one another Definition 1 Declaration of the properties of a new object, written only in terms of other well-defined objects and axioms Theorem A true statement in a theory which relates defined terms and other statements within the theory; Often stated in the form: If p, then q where p and q are statements which are true or false; Hypothesis(es)/ Conclusion 1 Definition versus Theorem Webster Proof and Its History 8/ / 1

46 Fundamental Terms II Proof Webster Proof and Its History 8/ / 1

47 Fundamental Terms II Proof A sequence of logically permissible steps connecting the hypothesis to the conclusion uses rules of inference and logic Webster Proof and Its History 8/ / 1

48 Fundamental Terms II Proof A sequence of logically permissible steps connecting the hypothesis to the conclusion uses rules of inference and logic Many types of proof, many proof techniques Webster Proof and Its History 8/ / 1

49 Fundamental Terms II Proof A sequence of logically permissible steps connecting the hypothesis to the conclusion uses rules of inference and logic Many types of proof, many proof techniques Lemma, Corollary, Proposition Webster Proof and Its History 8/ / 1

50 Fundamental Terms II Proof A sequence of logically permissible steps connecting the hypothesis to the conclusion uses rules of inference and logic Many types of proof, many proof techniques Lemma, Corollary, Proposition A Lemma is a small statement which only requires a small proof; usually a lemma and its proof are part of a larger proof of a theorem Webster Proof and Its History 8/ / 1

51 Fundamental Terms II Proof A sequence of logically permissible steps connecting the hypothesis to the conclusion uses rules of inference and logic Many types of proof, many proof techniques Lemma, Corollary, Proposition A Lemma is a small statement which only requires a small proof; usually a lemma and its proof are part of a larger proof of a theorem A Corollary is a small proposition which follows rather immediately from a theorem or its proof. Webster Proof and Its History 8/ / 1

52 Fundamental Terms II Proof A sequence of logically permissible steps connecting the hypothesis to the conclusion uses rules of inference and logic Many types of proof, many proof techniques Lemma, Corollary, Proposition A Lemma is a small statement which only requires a small proof; usually a lemma and its proof are part of a larger proof of a theorem A Corollary is a small proposition which follows rather immediately from a theorem or its proof. A Proposition is just a (quantifiable) statement which is true or false; usually does not require much proof Webster Proof and Its History 8/ / 1

53 Fundamental Notions I Modus ponens: Webster Proof and Its History 8/ / 1

54 Fundamental Notions I Modus ponens: If the statement p implies q is true, and the hypothesis p holds, then we know q must hold. Webster Proof and Its History 8/ / 1

55 Fundamental Notions I Modus ponens: If the statement p implies q is true, and the hypothesis p holds, then we know q must hold. Proofs generally take the form: Assume p (which may not always hold), derive the conclusion q. Webster Proof and Its History 8/ / 1

56 Fundamental Notions I Modus ponens: If the statement p implies q is true, and the hypothesis p holds, then we know q must hold. Proofs generally take the form: Assume p (which may not always hold), derive the conclusion q. Whenever p, then q. Webster Proof and Its History 8/ / 1

57 Fundamental Notions I Modus ponens: If the statement p implies q is true, and the hypothesis p holds, then we know q must hold. Proofs generally take the form: Assume p (which may not always hold), derive the conclusion q. Whenever p, then q. p is sufficient for q; Webster Proof and Its History 8/ / 1

58 Fundamental Notions I Modus ponens: If the statement p implies q is true, and the hypothesis p holds, then we know q must hold. Proofs generally take the form: Assume p (which may not always hold), derive the conclusion q. Whenever p, then q. p is sufficient for q; q is necessary for p. Webster Proof and Its History 8/ / 1

59 Fundamental Notions I Modus ponens: If the statement p implies q is true, and the hypothesis p holds, then we know q must hold. Proofs generally take the form: Assume p (which may not always hold), derive the conclusion q. Whenever p, then q. p is sufficient for q; q is necessary for p. Sets A set is any (unordered) collection of (mathematical) objects; Webster Proof and Its History 8/ / 1

60 Fundamental Notions I Modus ponens: If the statement p implies q is true, and the hypothesis p holds, then we know q must hold. Proofs generally take the form: Assume p (which may not always hold), derive the conclusion q. Whenever p, then q. p is sufficient for q; q is necessary for p. Sets A set is any (unordered) collection of (mathematical) objects; note one can have sets of sets (examples to follow) Webster Proof and Its History 8/ / 1

61 Fundamental Notions I Modus ponens: If the statement p implies q is true, and the hypothesis p holds, then we know q must hold. Proofs generally take the form: Assume p (which may not always hold), derive the conclusion q. Whenever p, then q. p is sufficient for q; q is necessary for p. Sets A set is any (unordered) collection of (mathematical) objects; note one can have sets of sets (examples to follow) Statements: An (open) statement: any mathematical statement; not necessarily true or false. Webster Proof and Its History 8/ / 1

62 Fundamental Notions I Modus ponens: If the statement p implies q is true, and the hypothesis p holds, then we know q must hold. Proofs generally take the form: Assume p (which may not always hold), derive the conclusion q. Whenever p, then q. p is sufficient for q; q is necessary for p. Sets A set is any (unordered) collection of (mathematical) objects; note one can have sets of sets (examples to follow) Statements: An (open) statement: any mathematical statement; not necessarily true or false. Propositions: A statement which is true or false, and can be proved or disproved. Webster Proof and Its History 8/ / 1

63 Fundamental Notions I Modus ponens: If the statement p implies q is true, and the hypothesis p holds, then we know q must hold. Proofs generally take the form: Assume p (which may not always hold), derive the conclusion q. Whenever p, then q. p is sufficient for q; q is necessary for p. Sets A set is any (unordered) collection of (mathematical) objects; note one can have sets of sets (examples to follow) Statements: An (open) statement: any mathematical statement; not necessarily true or false. Propositions: A statement which is true or false, and can be proved or disproved. Remember Godel... Webster Proof and Its History 8/ / 1

64 Sets I The empty set: Webster Proof and Its History 8/ / 1

65 Sets I The empty set: Lists Webster Proof and Its History 8/ / 1

66 Sets I The empty set: Lists {1, 2, 3} Webster Proof and Its History 8/ / 1

67 Sets I The empty set: Lists {1, 2, 3} {x, y, z, p, q, r} Webster Proof and Its History 8/ / 1

68 Sets I The empty set: Lists {1, 2, 3} {x, y, z, p, q, r} {+,,, /} Webster Proof and Its History 8/ / 1

69 Sets I The empty set: Lists {1, 2, 3} {x, y, z, p, q, r} {+,,, /} {sin(θ), cos(θ), tan(θ)} Webster Proof and Its History 8/ / 1

70 Sets I The empty set: Lists {1, 2, 3} {x, y, z, p, q, r} {+,,, /} {sin(θ), cos(θ), tan(θ)} Ellipses Webster Proof and Its History 8/ / 1

71 Sets I The empty set: Lists {1, 2, 3} {x, y, z, p, q, r} {+,,, /} {sin(θ), cos(θ), tan(θ)} Ellipses {1, 2, 3,...} {a, b, c,..., z} {1, 3, 5, 7,...} Webster Proof and Its History 8/ / 1

72 Sets I The empty set: Lists {1, 2, 3} {x, y, z, p, q, r} {+,,, /} {sin(θ), cos(θ), tan(θ)} Ellipses {1, 2, 3,...} {a, b, c,..., z} {1, 3, 5, 7,...} {2, 4,...} Webster Proof and Its History 8/ / 1

73 Sets I The empty set: Lists {1, 2, 3} {x, y, z, p, q, r} {+,,, /} {sin(θ), cos(θ), tan(θ)} Ellipses {1, 2, 3,...} {a, b, c,..., z} {1, 3, 5, 7,...} {2, 4,...} BAD Webster Proof and Its History 8/ / 1

74 Sets I The empty set: Lists {1, 2, 3} {x, y, z, p, q, r} {+,,, /} {sin(θ), cos(θ), tan(θ)} Ellipses {1, 2, 3,...} {a, b, c,..., z} {1, 3, 5, 7,...} {2, 4,...} BAD Set builder Webster Proof and Its History 8/ / 1

75 Sets I The empty set: Lists {1, 2, 3} {x, y, z, p, q, r} {+,,, /} {sin(θ), cos(θ), tan(θ)} Ellipses {1, 2, 3,...} {a, b, c,..., z} {1, 3, 5, 7,...} {2, 4,...} BAD Set builder {x : x > 0} {x x 2 + 5x + 6 = 0} {z z = 2 m + 1, where m is an integer} Webster Proof and Its History 8/ / 1

76 Sets II Subset/superset: Webster Proof and Its History 8/ / 1

77 Sets II Subset/superset: S R if all members of S are also present in R (proper subset: S R or S R) Webster Proof and Its History 8/ / 1

78 Sets II Subset/superset: S R if all members of S are also present in R (proper subset: S R or S R) Universe: Webster Proof and Its History 8/ / 1

79 Sets II Subset/superset: S R if all members of S are also present in R (proper subset: S R or S R) Universe: U is the set from which all sets in considerations are drawn; Webster Proof and Its History 8/ / 1

80 Sets II Subset/superset: S R if all members of S are also present in R (proper subset: S R or S R) Universe: U is the set from which all sets in considerations are drawn; if we are interested in a set S then S U (context dependent) Webster Proof and Its History 8/ / 1

81 Sets II Subset/superset: S R if all members of S are also present in R (proper subset: S R or S R) Universe: U is the set from which all sets in considerations are drawn; if we are interested in a set S then S U (context dependent) When talking about numbers, U usually equals C, R, Z or N Webster Proof and Its History 8/ / 1

82 Sets II Subset/superset: S R if all members of S are also present in R (proper subset: S R or S R) Universe: U is the set from which all sets in considerations are drawn; if we are interested in a set S then S U (context dependent) When talking about numbers, U usually equals C, R, Z or N Complement: For a set S U its complement is everything in U not in S Webster Proof and Its History 8/ / 1

83 Sets II Subset/superset: S R if all members of S are also present in R (proper subset: S R or S R) Universe: U is the set from which all sets in considerations are drawn; if we are interested in a set S then S U (context dependent) When talking about numbers, U usually equals C, R, Z or N Complement: For a set S U its complement is everything in U not in S S = U S = U \ S, where and \ are set minus (more later) Webster Proof and Its History 8/ / 1

84 Sets II Subset/superset: S R if all members of S are also present in R (proper subset: S R or S R) Universe: U is the set from which all sets in considerations are drawn; if we are interested in a set S then S U (context dependent) When talking about numbers, U usually equals C, R, Z or N Complement: For a set S U its complement is everything in U not in S S = U S = U \ S, where and \ are set minus (more later) Sometimes this is written as S C Webster Proof and Its History 8/ / 1

85 Sets II Subset/superset: S R if all members of S are also present in R (proper subset: S R or S R) Universe: U is the set from which all sets in considerations are drawn; if we are interested in a set S then S U (context dependent) When talking about numbers, U usually equals C, R, Z or N Complement: For a set S U its complement is everything in U not in S S = U S = U \ S, where and \ are set minus (more later) Sometimes this is written as S C x is an element of S is written x S, Webster Proof and Its History 8/ / 1

86 Sets II Subset/superset: S R if all members of S are also present in R (proper subset: S R or S R) Universe: U is the set from which all sets in considerations are drawn; if we are interested in a set S then S U (context dependent) When talking about numbers, U usually equals C, R, Z or N Complement: For a set S U its complement is everything in U not in S S = U S = U \ S, where and \ are set minus (more later) Sometimes this is written as S C x is an element of S is written x S, if x / S, then x S. Webster Proof and Its History 8/ / 1

87 Sets III Let S, R U. Set Equality: S = R if Webster Proof and Its History 8/ / 1

88 Sets III Let S, R U. Set Equality: S = R if S R and R S Webster Proof and Its History 8/ / 1

89 Sets III Let S, R U. Set Equality: S = R if S R and R S S and R have exactly the same elements Webster Proof and Its History 8/ / 1

90 Sets III Let S, R U. Set Equality: S = R if S R and R S S and R have exactly the same elements Union: Webster Proof and Its History 8/ / 1

91 Sets III Let S, R U. Set Equality: S = R if S R and R S S and R have exactly the same elements Union: S R {x U : x S or x R} Webster Proof and Its History 8/ / 1

92 Sets III Let S, R U. Set Equality: S = R if S R and R S S and R have exactly the same elements Union: S R {x U : x S or x R} Intersection: Webster Proof and Its History 8/ / 1

93 Sets III Let S, R U. Set Equality: S = R if S R and R S S and R have exactly the same elements Union: S R {x U : x S or x R} Intersection: S R {x U : x S and x R} Webster Proof and Its History 8/ / 1

94 Sets III Let S, R U. Set Equality: S = R if S R and R S S and R have exactly the same elements Union: S R {x U : x S or x R} Intersection: S R {x U : x S and x R} Difference: Webster Proof and Its History 8/ / 1

95 Sets III Let S, R U. Set Equality: S = R if S R and R S S and R have exactly the same elements Union: S R {x U : x S or x R} Intersection: S R {x U : x S and x R} Difference: S R = {x S : s / R} = S \ R Webster Proof and Its History 8/ / 1

96 Sets III Let S, R U. Set Equality: S = R if S R and R S S and R have exactly the same elements Union: S R {x U : x S or x R} Intersection: S R {x U : x S and x R} Difference: S R = {x S : s / R} = S \ R If S R then S R =... Webster Proof and Its History 8/ / 1

97 Sets III Let S, R U. Set Equality: S = R if S R and R S S and R have exactly the same elements Union: S R {x U : x S or x R} Intersection: S R {x U : x S and x R} Difference: S R = {x S : s / R} = S \ R If S R then S R =... Symmetric Difference: Webster Proof and Its History 8/ / 1

98 Sets III Let S, R U. Set Equality: S = R if S R and R S S and R have exactly the same elements Union: S R {x U : x S or x R} Intersection: S R {x U : x S and x R} Difference: S R = {x S : s / R} = S \ R If S R then S R =... Symmetric Difference: S R =... Webster Proof and Its History 8/ / 1

99 Fundamental Notions II Quantifiers (constructs to quantify statements) Webster Proof and Its History 8/ / 1

100 Fundamental Notions II Quantifiers (constructs to quantify statements) For all Webster Proof and Its History 8/ / 1

101 Fundamental Notions II Quantifiers (constructs to quantify statements) For all Webster Proof and Its History 8/ / 1

102 Fundamental Notions II Quantifiers (constructs to quantify statements) For all There exists Webster Proof and Its History 8/ / 1

103 Fundamental Notions II Quantifiers (constructs to quantify statements) For all There exists Webster Proof and Its History 8/ / 1

104 Fundamental Notions II Quantifiers (constructs to quantify statements) For all There exists There exists a unique Webster Proof and Its History 8/ / 1

105 Fundamental Notions II Quantifiers (constructs to quantify statements) For all There exists There exists a unique! Webster Proof and Its History 8/ / 1

106 Fundamental Notions II Quantifiers (constructs to quantify statements) For all There exists There exists a unique! Negation: For any statement P, the negation (read: not P ) is written: Webster Proof and Its History 8/ / 1

107 Fundamental Notions II Quantifiers (constructs to quantify statements) For all There exists There exists a unique! Negation: For any statement P, the negation (read: not P ) is written: P Webster Proof and Its History 8/ / 1

108 Fundamental Notions II Quantifiers (constructs to quantify statements) For all There exists There exists a unique! Negation: For any statement P, the negation (read: not P ) is written: P Cardinality: Webster Proof and Its History 8/ / 1

109 Fundamental Notions II Quantifiers (constructs to quantify statements) For all There exists There exists a unique! Negation: For any statement P, the negation (read: not P ) is written: P Cardinality: S = #(S) number of elements in S. Webster Proof and Its History 8/ / 1

110 Fundamental Notions II Quantifiers (constructs to quantify statements) For all There exists There exists a unique! Negation: For any statement P, the negation (read: not P ) is written: P Cardinality: S = #(S) number of elements in S. Infinity: Webster Proof and Its History 8/ / 1

111 Fundamental Notions II Quantifiers (constructs to quantify statements) For all There exists There exists a unique! Negation: For any statement P, the negation (read: not P ) is written: P Cardinality: S = #(S) number of elements in S. Infinity: Concept (not to be treated like a number); Webster Proof and Its History 8/ / 1

112 Fundamental Notions II Quantifiers (constructs to quantify statements) For all There exists There exists a unique! Negation: For any statement P, the negation (read: not P ) is written: P Cardinality: S = #(S) number of elements in S. Infinity: Concept (not to be treated like a number); a class of cardinalities of certain sets; Webster Proof and Its History 8/ / 1

113 Fundamental Notions II Quantifiers (constructs to quantify statements) For all There exists There exists a unique! Negation: For any statement P, the negation (read: not P ) is written: P Cardinality: S = #(S) number of elements in S. Infinity: Concept (not to be treated like a number); a class of cardinalities of certain sets; not finite Webster Proof and Its History 8/ / 1