Indicator functionFrom Wikipedia, the free encyclopediaContents1 Choice function 11.1 An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 History and importance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Renement of the notion of choice function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.1 Bourbaki tau function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Disjoint union 32.1 Example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Set theory denition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.3 Category theory point of view . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Indicator function 53.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.2 Remark on notation and terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.3 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.4 Mean, variance and covariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.5 Characteristic function in recursion theory, Gdels and Kleenes representing function . . . . . . . 73.6 Characteristic function in fuzzy set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.7 Derivatives of the indicator function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 List of rules of inference 104.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.2 Rules for classical sentential calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.2.1 Rules for negations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.2.2 Rules for conditionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11iii CONTENTS4.2.3 Rules for conjunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.2.4 Rules for disjunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.2.5 Rules for biconditionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.3 Rules of classical predicate calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.4 Table: Rules of Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.4.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.4.2 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 Pointed set 165.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175.4 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 185.4.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185.4.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185.4.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18Chapter 1Choice functionFor the combinatorial choice function C(n, k), see Combination and Binomial coecient.Achoice function (selector, selection) is a mathematical function f that is dened on some collection X of nonemptysets and assigns to each set S in that collection some element f(S) of S. In other words, f is a choice function for X ifand only if it belongs to the direct product of X.1.1 An exampleLet X = { {1,4,7}, {9}, {2,7} }. Then the function that assigns 7 to the set {1,4,7}, 9 to {9}, and 2 to {2,7} is achoice function on X.1.2 History and importanceErnst Zermelo (1904) introduced choice functions as well as the axiom of choice (AC) and proved the well-orderingtheorem,[1] which states that every set can be well-ordered. AC states that every set of nonempty sets has a choicefunction. A weaker form of AC, the axiom of countable choice (AC) states that every countable set of nonemptysets has a choice function. However, in the absence of either AC or AC, some sets can still be shown to have achoice function.If X is a nite set of nonempty sets, then one can construct a choice function for X by picking one elementfrom each member of X. This requires only nitely many choices, so neither AC or AC is needed.If every member of X is a nonempty set, and the unionX is well-ordered, then one may choose the leastelement of each member of X . In this case, it was possible to simultaneously well-order every member of Xby making just one choice of a well-order of the union, so neither AC nor AC was needed. (This exampleshows that the well-ordering theorem implies AC. The converse is also true, but less trivial.)1.3 Renement of the notion of choice functionA function f:A B is said to be a selection of a multivalued map :A B (that is, a function :A P(B)from A to the power set P(B) ), ifa A(f(a) (a)) .The existence of more regular choice functions, namely continuous or measurable selections is important in the theoryof dierential inclusions, optimal control, and mathematical economics.[2]12 CHAPTER 1. CHOICE FUNCTION1.3.1 Bourbaki tau functionNicolas Bourbaki used epsilon calculus for their foundations that had a symbol that could be interpreted as choosingan object (if one existed) that satises a given proposition.So if P(x) is a predicate, then x(P) is the object thatsatises P (if one exists, otherwise it returns an arbitrary object). Hence we may obtain quantiers from the choicefunction, for example P(x(P)) was equivalent to (x)(P(x)) .[3]However, Bourbakis choice operator is stronger than usual: its a global choice operator. That is, it implies the axiomof global choice.[4] Hilbert realized this when introducing epsilon calculus.[5]1.4 See alsoAxiom of countable choiceHausdor paradoxHemicontinuity1.5 Notes[1] Zermelo, Ernst (1904). Beweis, dass jede Menge wohlgeordnet werden kann.Mathematische Annalen 59 (4): 51416.doi:10.1007/BF01445300.[2] Border, Kim C. (1989). Fixed Point Theorems with Applications to Economics and Game Theory. Cambridge UniversityPress. ISBN 0-521-26564-9.[3] Bourbaki, Nicolas. Elements of Mathematics: Theory of Sets. ISBN 0-201-00634-0.[4] John Harrison, The Bourbaki View eprint.[5] Here, moreover, we come upon a very remarkable circumstance, namely, that all of these transnite axioms are derivablefrom a single axiom, one that also contains the core of one of the most attacked axioms in the literature of mathematics,namely, the axiom of choice: A(a) A((A)) , where is the transnite logical choice function. Hilbert (1925), Onthe Innite, excerpted in Jean van Heijenoort, From Frege to Gdel, p. 382. From nCatLab.1.6 ReferencesThis article incorporates material from Choice function on PlanetMath, which is licensed under the Creative CommonsAttribution/Share-Alike License.Chapter 2Disjoint unionIn set theory, the disjoint union (or discriminated union) of a family of sets is a modied union operation thatindexes the elements according to which set they originated in. Or slightly dierent from this, the disjoint union ofa family of subsets is the usual union of the subsets which are pairwise disjoint disjoint sets means they have noelement in common.Note that these two concepts are dierent but strongly related. Moreover, it seems that they are essentially identicalto each other in category theory. That is, both are realizations of the coproduct of category of sets.2.1 ExampleDisjoint union of sets A0 = {1, 2, 3} and A1 = {1, 2} can be computed by nding:A0= {(1, 0), (2, 0), (3, 0)}A1= {(1, 1), (2, 1)}soA0 A1= A0 A1= {(1, 0), (2, 0), (3, 0), (1, 1), (2, 1)}2.2 Set theory denitionFormally, let {Ai : i I} be a family of sets indexed by i. The disjoint union of this family is the setiIAi=iI{(x, i) : x Ai}.The elements of the disjoint union are ordered pairs (x, i). Here i serves as an auxiliary index that indicates which Aithe element x came from.Each of the sets Ai is canonically isomorphic to the setAi= {(x, i) : x Ai}.Through this isomorphism, one may consider that Ai is canonically embedded in the disjoint union. For i j, the setsAi* and Aj* are disjoint even if the sets Ai and Aj are not.In the extreme case where each of the Ai is equal to some xed set A for each i I, the disjoint union is the Cartesianproduct of A and I:34 CHAPTER 2. DISJOINT UNIONiIAi= AI.One may occasionally see the notationiIAifor the disjoint union of a family of sets, or the notation A + B for the disjoint union of two sets. This notation ismeant to be suggestive of the fact that the cardinality of the disjoint union is the sum of the cardinalities of the termsin the family. Compare this to the notation for the Cartesian product of a family of sets.Disjoint unions are also sometimes writteniI Ai or iI Ai .In the language of category theory, the disjoint union is the coproduct in the category of sets. It therefore satises theassociated universal property. This also means that the disjoint union is the categorical dual of the Cartesian productconstruction. See coproduct for more details.For many purposes, the particular choice of auxiliary index is unimportant, and in a simplifying abuse of notation,the indexed family can be treated simply as a collection of sets. In this case Ai is referred to as a copy of Ai and thenotation ACA is sometimes used.2.3 Category theory point of viewIn category theory the disjoint union is dened as a coproduct in the category of sets.As such, the disjoint union is dened up to an isomorphism, and the above denition is just one realization of thecoproduct, among others. When the sets are pairwise disjoint, the usual union is another realization of the coproduct.This justies the second denition in the lead.This categorical aspect of the disjoint union explains whyis frequently used, instead of, to denote coproduct.2.4 See alsoCoproductDisjoint union (topology)Disjoint union of graphsPartition of a setSum typeTagged unionUnion (computer science)2.5 ReferencesLang, Serge (2004), Algebra, Graduate Texts in Mathematics 211 (Corrected fourth printing, revised thirded.), New York: Springer-Verlag, p. 60, ISBN 978-0-387-95385-4Weisstein, Eric W., Disjoint Union, MathWorld.Chapter 3Indicator functionThe graph of the indicator function of a two-dimensional subset of a square.In mathematics, an indicator function or a characteristic function is a function dened on a set X that indicatesmembership of an element in a subset A of X, having the value 1 for all elements of A and the value 0 for all elementsof X not in A. It is usually denoted by a bold or blackboard bold 1 symbol with a subscript describing the event ofinclusion.3.1 DenitionThe indicator function of a subset A of a set X is a function1A:X {0, 1}dened as56 CHAPTER 3. INDICATOR FUNCTION1A(x) :={1 ifx A,0 ifx/ A.The Iverson bracket allows the equivalent notation, [x A] , to be used instead of 1A(x) .The function 1A is sometimes denoted IA , A or even just A . (The Greek letter appears because it is the initialletter of the Greek word characteristic.)3.2 Remark on notation and terminologyThe notation 1A is also used to denote the identity function of A.The notation A is also used to denote the characteristic function in convex analysis.A related concept in statistics is that of a dummy variable.(This must not be confused with dummy variables asthat term is usually used in mathematics, also called a bound variable.)The term "characteristic function" has an unrelated meaning in probability theory. For this reason, probabilists usethe term indicator function for the function dened here almost exclusively, while mathematicians in other elds aremore likely to use the term characteristic function to describe the function that indicates membership in a set.3.3 Basic propertiesThe indicator or characteristic function of a subset A of some set X, maps elements of X to the range {0,1}.This mapping is surjective only when A is a non-empty proper subset of X. If A X, then 1A = 1. By a similarargument, if A then 1A = 0.In the following, the dot represents multiplication, 11 = 1, 10 = 0 etc. "+" and "" represent addition and subtraction." " and " " is intersection and union, respectively.If A and B are two subsets of X , then1AB= min{1A, 1B} = 1A 1B,1AB= max{1A, 1B} = 1A + 1B 1A 1B,and the indicator function of the complement of A i.e.ACis:1A= 1 1AMore generally, suppose A1, . . . , An is a collection of subsets of X. For any x X:kI(1 1Ak(x))is clearly a product of 0s and 1s. This product has the value 1 at precisely those x X that belong to none of the setsAk and is 0 otherwise. That iskI(1 1Ak) = 1XkAk= 1 1kAk.Expanding the product on the left hand side,3.4. MEAN, VARIANCE AND COVARIANCE 71kAk= 1 F{1,2,...,n}(1)|F|1FAk= =F{1,2,...,n}(1)|F|+11FAkwhere |F| is the cardinality of F. This is one form of the principle of inclusion-exclusion.As suggested by the previous example, the indicator function is a useful notational device in combinatorics. Thenotation is used in other places as well, for instance in probability theory: if X is a probability space with probabilitymeasure P andA is a measurable set, then 1A becomes a random variable whose expected value is equal to theprobability of A :E(1A) =X1A(x) dP =AdP = P(A)This identity is used in a simple proof of Markovs inequality.In many cases, such as order theory, the inverse of the indicator function may be dened. This is commonly calledthe generalized Mbius function, as a generalization of the inverse of the indicator function in elementary numbertheory, the Mbius function. (See paragraph below about the use of the inverse in classical recursion theory.)3.4 Mean, variance and covarianceGiven a probability space (, F, P) with A F , the indicator randomvariable 1A: Ris dened by 1A() = 1if A, otherwise 1A() = 0.MeanE(1A()) = P(A)VarianceVar(1A()) = P(A)(1 P(A))CovarianceCov(1A(), 1B()) = P(A B) P(A) P(B)3.5 Characteristic function in recursion theory, Gdels and Kleenes rep-resenting functionKurt Gdel described the representing function in his 1934 paper On Undecidable Propositions of Formal Mathe-matical Systems. (The paper appears on pp. 4174 in Martin Davis ed. The Undecidable):There shall correspond to each class or relation R a representing function (x1, . . ., x) = 0 if R(x1, .. ., x) and (x1, . . ., x) = 1 if ~R(x1, . . ., x). (p. 42; the "~" indicates logical inversion i.e. NOT)Stephen Kleene (1952) (p. 227) oers up the same denition in the context of the primitive recursive functions as afunction of a predicate P takes on values 0 if the predicate is true and 1 if the predicate is false.For example, because the product of characteristic functions 1*2* . . . * = 0 whenever any one of the functionsequals 0, it plays the role of logical OR: IF 1 = 0 OR 2 = 0 OR . . . OR = 0 THEN their product is 0. Whatappears to the modern reader as the representing functions logical inversion, i.e. the representing function is 0 whenthe function R is true or satised, plays a useful role in Kleenes denition of the logical functions OR, AND, andIMPLY (p.228), the bounded- (p.228) and unbounded- (p.279) mu operators (Kleene (1952)) and the CASEfunction (p. 229).3.6 Characteristic function in fuzzy set theoryIn classical mathematics, characteristic functions of sets only take values 1 (members) or 0 (non-members). In fuzzyset theory, characteristic functions are generalized to take value in the real unit interval [0, 1], or more generally, in8 CHAPTER 3. INDICATOR FUNCTIONsome algebra or structure (usually required to be at least a poset or lattice). Such generalized characteristic functionsare more usually called membership functions, and the corresponding sets are called fuzzy sets. Fuzzy sets modelthe gradual change in the membership degree seen in many real-world predicates like tall, warm, etc.3.7 Derivatives of the indicator functionA particular indicator function, which is very well known, is the Heaviside step function. The Heaviside step functionis the indicator function of the one-dimensional positive half-line, i.e. the domain [0, ). It is well known that thedistributional derivative of the Heaviside step function, indicated by H(x), is equal to the Dirac delta function, i.e.(x) =dH(x)dx,with the following property:f(x) (x)dx = f(0).The derivative of the Heaviside step function can be seen as the 'inward normal derivative' at the 'boundary' of thedomain given by the positive half-line. In higher dimensions, the derivative naturally generalises to the inward normalderivative, while the Heaviside step function naturally generalises to the indicator function of some domain D. Thesurface of D will be denoted by S. Proceeding, it can be derived that the inward normal derivative of the indicatorgives rise to a 'surface delta function', which can be indicated by S(x):S(x) = nx x1xDwhere n is the outward normal of the surface S. This 'surface delta function' has the following property:[1]Rnf(x) nx x1xDdnx =

Sf() dn1.By setting the function f equal to one, it follows that the inward normal derivative of the indicator integrates to thenumerical value of the surface area S.3.8 See alsoDirac measureLaplacian of the indicatorDirac deltaExtension (predicate logic)Free variables and bound variablesHeaviside step functionIverson bracketKronecker delta, a function that can be viewed as an indicator for the identity relationMacaulay bracketsMultisetMembership function3.9. NOTES 9Simple functionDummy variable (statistics)Statistical classicationZero-one loss function3.9 Notes[1] Lange, Rutger-Jan (2012), Potential theory, path integrals and the Laplacian of the indicator, Journal of High EnergyPhysics (Springer) 2012 (11): 2930, arXiv:1302.0864, Bibcode:2012JHEP...11..032L, doi:10.1007/JHEP11(2012)0323.10 ReferencesFolland, G.B. (1999). Real Analysis: Modern Techniques and Their Applications (Second ed.). John Wiley &Sons, Inc.Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L.; Stein, Cliord (2001). Section 5.2: Indicatorrandom variables.Introduction to Algorithms (Second ed.). MIT Press and McGraw-Hill. pp. 9499. ISBN0-262-03293-7.Davis, Martin, ed. (1965). The Undecidable. New York: Raven Press Books, Ltd.Kleene, Stephen (1971) [1952]. Introduction to Metamathematics (Sixth Reprint with corrections). Nether-lands: Wolters-Noordho Publishing and North Holland Publishing Company.Boolos, George; Burgess, John P.; Jerey, Richard C. (2002). Computability and Logic. Cambridge UK:Cambridge University Press. ISBN 0-521-00758-5.Zadeh, Lot A. (June 1965). Fuzzy sets (PDF). Information and Control 8 (3): 338353. doi:10.1016/S0019-9958(65)90241-X.Goguen, Joseph (1967). "L-fuzzy sets. Journal of Mathematical Analysis and Applications 18 (1): 145174.doi:10.1016/0022-247X(67)90189-8.Chapter 4List of rules of inferenceThis is a list of rules of inference, logical laws that relate to mathematical formulae.4.1 IntroductionRules of inference are syntactical transform rules which one can use to infer a conclusion from a premise to createan argument. A set of rules can be used to infer any valid conclusion if it is complete, while never inferring an invalidconclusion, if it is sound. A sound and complete set of rules need not include every rule in the following list, as manyof the rules are redundant, and can be proven with the other rules.Discharge rules permit inference from a subderivation based on a temporary assumption. Below, the notation indicates such a subderivation from the temporary assumption to .4.2 Rules for classical sentential calculusSentential calculus is also known as propositional calculus.4.2.1 Rules for negationsReductio ad absurdum (or Negation Introduction) Reductio ad absurdum (related to the law of excluded middle) Noncontradiction (or Negation Elimination)104.2. RULES FOR CLASSICAL SENTENTIAL CALCULUS 11Double negation eliminationDouble negation introduction4.2.2 Rules for conditionalsDeduction theorem (or Conditional Introduction) Modus ponens (or Conditional Elimination) Modus tollens 4.2.3 Rules for conjunctionsAdjunction (or Conjunction Introduction) Simplication (or Conjunction Elimination) 12 CHAPTER 4. LIST OF RULES OF INFERENCE4.2.4 Rules for disjunctionsAddition (or Disjunction Introduction) Case analysis (or Proof by Cases or Argument by Cases) Disjunctive syllogism Constructive dilemma 4.2.5 Rules for biconditionalsBiconditional introduction Biconditional Elimination 4.3. RULES OF CLASSICAL PREDICATE CALCULUS 13 4.3 Rules of classical predicate calculusIn the following rules, (/) is exactly like except for having the term everywhere has the free variable .Universal Generalization (or Universal Introduction)(/)Restriction 1: is a variable which does not occur in .Restriction 2: is not mentioned in any hypothesis or undischarged assumptions.Universal Instantiation (or Universal Elimination)(/)Restriction: No free occurrence of in falls within the scope of a quantier quantifying a variable occurring in .Existential Generalization (or Existential Introduction)(/)Restriction: No free occurrence of in falls within the scope of a quantier quantifying a variable occurring in .Existential Instantiation (or Existential Elimination)(/) Restriction 1: is a variable which does not occur in .Restriction 2: There is no occurrence, free or bound, of in .Restriction 3: is not mentioned in any hypothesis or undischarged assumptions.14 CHAPTER 4. LIST OF RULES OF INFERENCE4.4 Table: Rules of InferenceThe rules above can be summed up in the following table.[1] The "Tautology" column shows how to interpret thenotation of a given rule.All rules use the basic logic operators. A complete table of logic operators is shown by a truth table, giving deni-tions of all the possible (16) truth functions of 2 boolean variables (p, q):where T = true and F = false, and, the columns are the logical operators: 0, false, Contradiction; 1, NOR, LogicalNOR; 2, Converse nonimplication; 3, p, Negation; 4, Material nonimplication; 5, q, Negation; 6, XOR, Exclusivedisjunction;7, NAND, Logical NAND; 8, AND, Logical conjunction;9, XNOR, If and only if, Logical bicondi-tional; 10, q, Projection function; 11, if/then, Logical implication; 12, p, Projection function; 13, then/if, Converseimplication; 14, OR, Logical disjunction; 15, true, Tautology.Each logic operator can be used in an assertion about variables and operations, showing a basic rule of inference.Examples:The column-14 operator (OR), shows Addition rule: when p=T (the hypothesis selects the rst two lines of thetable), we see (at column-14) that pq=T.We can see also that, with the same premise, another conclusions are valid: columns 12, 14 and 15are T.The column-8 operator (AND), shows Simplication rule: when pq=T (rst line of the table), we see thatp=T.With this premise, we also conclude that q=T, pq=T, etc. as showed by columns 9-15.The column-11 operator (IF/THEN), shows Modus ponens rule: when pq=T and p=T only one line of thetruth table (the rst) satises these two conditions. On this line, q is also true. Therefore, whenever p q istrue and p is true, q must also be true.Machines and well-trained people use this look at table approach to do basic inferences, and to check if other infer-ences (for the same premises) can be obtained.4.4.1 Example 1Let us consider the following assumptions: If it rains today, then we will not go on a canoe today. If we do not goon a canoe trip today, then we will go on a canoe trip tomorrow. Therefore (Mathematical symbol for therefore is ), if it rains today, we will go on a canoe trip tomorrow. To make use of the rules of inference in the above tablewe let p be the proposition If it rains today, q be We will not go on a canoe today and let r be We will go on acanoe trip tomorrow. Then this argument is of the form:p qq r p r4.4.2 Example 2Let us consider a more complex set of assumptions: It is not sunny today and it is colder than yesterday. We willgo swimming only if it is sunny, If we do not go swimming, then we will have a barbecue, and If we will havea barbecue, then we will be home by sunset lead to the conclusion We will be home by sunset. Proof by rules ofinference: Let p be the proposition It is sunny today, q the proposition It is colder than yesterday, r the propositionWe will go swimming, s the proposition We will have a barbecue, and t the proposition We will be home bysunset. Then the hypotheses become p q, r p, r s and s t . Using our intuition we conjecture thatthe conclusion might be t . Using the Rules of Inference table we can proof the conjecture easily:4.5. REFERENCES 154.5 References[1] Kenneth H. Rosen: Discrete Mathematics and its Applications, Fifth Edition, p. 58.Chapter 5Pointed setIn mathematics, a pointed set[1][2] (also based set[1] or rooted set[3]) is an ordered pair (X, x0) where X is a setand x0 is an element of X called the base point,[2] also spelled basepoint.[4]:1011Maps between pointed sets(X, x0) and(Y, y0) (calledbasedmaps,[5]pointedmaps,[4] orpoint-preservingmaps[6]) are functions from X to Y that map one basepoint to another, i.e. a map f: X Y such that f(x0) = y0. This is usually denotedf: (X, x0) (Y, y0)Pointed sets may be regarded as a rather simple algebraic structure. In the sense of universal algebra, they arestructures with a single nullary operation which picks out the basepoint.[7]The class of all pointed sets together with the class of all based maps form a category. In this category the pointedsingleton set ({a}, a) is an initial object and a terminal object,[1] i.e. a zero object.[4]:226 There is a faithful functorfrom usual sets to pointed sets, but it is not full and these categories are not equivalent.[8]:44 In particular, the emptyset is not a pointed set, for it has no element that can be chosen as base point.[9]The category of pointed sets and based maps is equivalent to but not isomorphic with the category of sets and partialfunctions.[6] One textbook notes that This formal completion of sets and partial maps by adding improper, in-nite elements was reinvented many times, in particular, in topology (one-point compactication) and in theoreticalcomputer science.[10]The category of pointed sets and pointed maps is isomorphic to the co-slice category 1 Set , where 1 is a singletonset.[8]:46[11]The category of pointed sets and pointed maps has both products and co-products, but it is not a distributive cate-gory.[9]Many algebraic structures are pointed sets in a rather trivial way. For example, groups are pointed sets by choosing theidentity element as the basepoint, so that group homomorphisms are point-preserving maps.[12]:24 This observationcan be restated in category theoretic terms as the existence of a forgetful functor from groups to pointed sets.[12]:582A pointed set may be seen as a pointed space under the discrete topology or as a vector space over the eld with oneelement.[13]As rooted set the notion naturally appears in the study of antimatroids[3] and transportation polytopes.[14]5.1 See alsoAccessible pointed graph5.2 References[1] Mac Lane (1998) p.26165.3. EXTERNAL LINKS 17[2] Grgory Berhuy (2010). An Introduction to Galois Cohomology and Its Applications. London Mathematical Society LectureNote Series 377. Cambridge University Press. p. 34. ISBN 0-521-73866-0. Zbl 1207.12003.[3] Korte, Bernhard; Lovsz, Lszl; Schrader, Rainer (1991), Greedoids, Algorithms and Combinatorics 4, NewYork, Berlin:Springer-Verlag, chapter 3, ISBN 3-540-18190-3, Zbl 0733.05023[4] Joseph Rotman (2008). An Introduction to Homological Algebra (2nd ed.). Springer Science & Business Media. ISBN978-0-387-68324-9.[5] Maunder, C. R. F. (1996), Algebraic Topology, Dover, p. 31.[6] Lutz Schrder (2001). Categories: a free tour. In Jrgen Koslowski and Austin Melton. Categorical Perspectives. SpringerScience & Business Media. p. 10. ISBN 978-0-8176-4186-3.[7] Saunders Mac Lane; Garrett Birkho (1999) [1988]. Algebra (3rd ed.). American Mathematical Soc. p. 497. ISBN978-0-8218-1646-2.[8] J. Adamek, H. Herrlich, G. Stecker, (18th January 2005) Abstract and Concrete Categories-The Joy of Cats[9] F. W. Lawvere; Stephen Hoel Schanuel (2009). Conceptual Mathematics: A First Introduction to Categories (2nd ed.).Cambridge University Press. pp. 296298. ISBN 978-0-521-89485-2.[10] Neal Koblitz; B. Zilber; Yu.I. Manin (2009). A Course in Mathematical Logic for Mathematicians.Springer Science &Business Media. p. 290. ISBN 978-1-4419-0615-1.[11] Francis Borceux; Dominique Bourn (2004). Mal'cev, Protomodular, Homological and Semi-Abelian Categories. SpringerScience & Business Media. p. 131. ISBN 978-1-4020-1961-6.[12] Paolo Alu (2009). Algebra: Chapter 0. American Mathematical Soc. ISBN 978-0-8218-4781-7.[13] Haran, M. J. Shai (2007), Non-additive geometry (PDF), Compositio Mathematica 143 (3): 618688, MR 2330442. Onp. 622, Haran writes We consider F -vector spaces as nite sets X with a distinguished zero element...[14] Klee, V.; Witzgall, C. (1970) [1968]. Facets and vertices of transportation polytopes. In George Bernard Dantzig.Mathematics of the Decision Sciences. Part 1. American Mathematical Soc. ISBN 0-8218-1111-0. OCLC 859802521.Mac Lane, Saunders (1998). Categories for the Working Mathematician (2nd ed.). Springer-Verlag. ISBN0-387-98403-8. Zbl 0906.18001.5.3 External linkshttp://mathoverflow.net/questions/22036/pullbacks-in-category-of-sets-and-partial-functionsPointed Set at PlanetMathhttp://ncatlab.org/nlab/show/pointed+object18 CHAPTER 5. POINTED SET5.4 Text and image sources, contributors, and licenses5.4.1 Text Choice functionSource: https://en.wikipedia.org/wiki/Choice_function?oldid=653509140 Contributors: Charles Matthews, Tobias Berge-mann, Giftlite, Chinasaur, Paul August, Rjwilmsi, Salix alba, RussBot, Piet Delport, Rick Norwood, Dreadstar, Vina-iwbot~enwiki, JR-Spriggs, Vanisaac, CRGreathouse, Myasuda, Kilva, NBeale, Odexios, Anonymous Dissident, Dreamafter, Megaloxantha, Randomblue,Squirreljml, Pqnelson, Addbot, DOI bot, Neodop, FrescoBot, Citation bot 1, 777sms, DrBizarro, EmausBot, Helpful Pixie Bot, BG19botand Anonymous: 14 Disjoint union Source: https://en.wikipedia.org/wiki/Disjoint_union?oldid=664597191 Contributors: Mav, Zundark, Tarquin, Takuya-Murata, Charles Matthews, Dcoetzee, Altenmann, MathMartin, Giftlite, Fropu, CyborgTosser, Abdull, Paul August, Oleg Alexandrov,Linas, Jacobolus, Salix alba, Mathbot, YurikBot, Hairy Dude, Bhny, Tong~enwiki, David Pierce, SmackBot, RDBury, Melchoir, Alan Mc-Beth, Nbarth, Wdvorak, Mets501, Thijs!bot, JAnDbot, Albmont, Americanhero, Pavel Jelnek, The enemies of god, PatHayes, Jamelan,Niceguyedc, Addbot, Arketyp, Gtgith, Vorbeigehende, Yodigo, Erik9bot, Stpasha, WBielas, WikitanvirBot, D.Lazard, ChuispastonBot,Akseli.palen, 10k, Freeze S, Deltahedron, Ktzell29 and Anonymous: 19 Indicator function Source: https://en.wikipedia.org/wiki/Indicator_function?oldid=674692770 Contributors: AxelBoldt, The Anome,Michael Hardy, Pcb21, Jordi Burguet Castell, Charles Matthews, Dcoetzee, Phil Boswell, Robbot, MathMartin, Centrx, Giftlite, Pi-otrus, CSTAR, Gauss, Icairns, Mormegil, ArnoldReinhold, Martpol, Paul August, Obradovic Goran, Oleg Alexandrov, Linas, Qwertyus,Rjwilmsi, Salix alba, Small potato, NickBush24, Trovatore, DavidHouse~enwiki, Banus, Bo Jacoby, SmackBot, BiT, Chris the speller,Octahedron80, Jon Awbrey, Vina-iwbot~enwiki, Wvbailey, Mets501, Eassin, Raystorm, CmdrObot, Thijs!bot, Helgus, Sherbrooke,Rosh3000, JAnDbot, Albmont, Sullivan.t.j, David Eppstein, Falcor84, MartinBot, Xetrov, VolkovBot, TXiKiBoT, Digby Tantrum, LBe-hounek, AlleborgoBot, Quietbritishjim, BotMultichill, Paintman, Melcombe, Mpd1989, UKoch, Colinvella, DragonBot, Addbot, Fg-nievinski, PV=nRT, Luckas-bot, Ht686rg90, AnomieBOT, Xqbot, TechBot, RibotBOT, Night Jaguar, Straightontillmorning, Maschen,Zfeinst, Bibcode Bot, BG19bot, Dlituiev, BattyBot, Dexbot, Rutgerjanlange, Salspaugh, DarenCline and Anonymous: 30 List of rules of inference Source: https://en.wikipedia.org/wiki/List_of_rules_of_inference?oldid=669311411 Contributors: Toby Bar-tels, Humanoid, AugPi, Poor Yorick, Jitse Niesen, Bkell, Jiy, ArnoldReinhold, ZeroOne, El C, Mathbot, JMRyan, Shadro, Mhss, ColoniesChris, Sidmow, Syrcatbot, Stwalkerster, JRSpriggs, CBM, Gregbard, Julian Mendez, Fireice, Magioladitis, Jonathanzung, Rezarob,Anonymous Dissident, Kumioko (renamed), Marc van Leeuwen, Jarble, Yobot, LilHelpa, Mark Renier, Momergil, JamesMazur22, In-nity ive, ClueBot NG, Mogism, TheKing44, YiFeiBot, Paul Er Coranjeh and Anonymous: 30 Pointed set Source: https://en.wikipedia.org/wiki/Pointed_set?oldid=657728392 Contributors: Revolver, Charles Matthews, Tobias Berge-mann, Fropu, Oleg Alexandrov, Salix alba, Benja, SmackBot, BiT, MrDomino, Mathsci, Konradek, David Eppstein, Julian I Do Stu,Classicalecon, Addbot, Yobot, AnomieBOT, Erik9bot, ZroBot, Zarboublian, Helpful Pixie Bot, Deltahedron, Tautological tau, JMPEAX, Broido and Anonymous: 45.4.2 Images File:Indicator_function_illustration.png Source: https://upload.wikimedia.org/wikipedia/commons/f/f5/Indicator_function_illustration.png License: Public domain Contributors: self-made with MATLAB, source code below Original artist: Oleg Alexandrov File:Logic_portal.svg Source: https://upload.wikimedia.org/wikipedia/commons/7/7c/Logic_portal.svg License: CC BY-SA 3.0 Con-tributors: Own work Original artist: Watchduck (a.k.a. Tilman Piesk) File:Question_book-new.svg Source: https://upload.wikimedia.org/wikipedia/en/9/99/Question_book-new.svg License: Cc-by-sa-3.0Contributors:Created from scratch in Adobe Illustrator. Based on Image:Question book.png created by User:Equazcion Original artist:Tkgd2007 File:Text_document_with_red_question_mark.svg Source: https://upload.wikimedia.org/wikipedia/commons/a/a4/Text_document_with_red_question_mark.svg License:Public domain Contributors:Created by bdesham with Inkscape; based upon Text-x-generic.svgfrom the Tango project. Original artist: Benjamin D. Esham (bdesham)5.4.3 Content license Creative Commons Attribution-Share Alike 3.0