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What is the difference between "if" and "iff" in formal logic?
Michael Scalise
if is an expression, while iff is a function often with multiple choices.
nycole4
Keara O'Conner
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What are the equality axioms?
Say there's a relation ~ between the two objects a and b such that a ~ b. We call ~ an equivalence relation if:i) a ~ a.ii) If a ~ b. then b ~ a.iii) If a ~ b and b ~ c, then a ~ c.Where c is another object.The three properties above are called the reflexive, symmetric, and transitive properties. Were those three properties all that was needed to define the equalityrelation, we could safely call them axioms. However, one more property is needed first. To show you why, I'll give an example.Consider the relation, "is parallel to," represented by . We'll check the properties above to see if is an equivalence relation.i) a a.Believe it or not, whether this statement is true is an ongoing debate. Many people feel that the parallel relation isn't defined for just one line, because it's a comparison. Well, if that were true, then you would have to say the same thing for everybinary equivalence relation; e.g., a triangle couldn't be similar to itself, or, even more preposterously, the statement a = a would have to be tossed out the window too. But, just to be formal, we'll use the following definition for parallel lines:Two lines are not parallel if they have exactlyone point in common; otherwise they are parallel.So, with that definition in hand, i) holds for .ii) If a b, then b a. True.iii) If a b and b c, then a c. True.Thus, the relation is an equivalence relation, but two parallel lines certainly don't have to be equal! So, we need an additional property to describe an equality relation:iv) If a ~ b and b ~ a, then a = b.Let's check iv) and see if this works for our relation :If a b and b a, then a = b. False. But, does it hold for the equality relation?If a = b and b = a then a = b. True. This is what's known as the antisymmetricproperty, and is what distinguishes equality from equivalence.But wait, we have a problem. We used the relation = in one of our "axioms" of equality. That doesn't work, because equality wasn't part of the signature of the formal languagewe're using here. By the way, the signature of the formal language that we are using is ~. So, any other non-logical symbol we use has to either be defined, or derived from axioms.Well, we have three possible ways out of this. We can either:1) Figure out a way to axiomize the = relation through the use of the ~ relation.2) Define the = relation.3) Add = to our language's signature.Well, 1) is not possible without the use of sets, and since the existence of sets isn't part of our signature either, we'd have to define a set, or add it to our language. This isn't very hard to do, but I'm not going to bother, because the result is what we're going to obtain from 2).Anyways, speaking of 2), let's define =.For all predicates (also called properties) P, and for all a and b, P(a) if and only if P(b) implies that a = b.In other words, for a to be equal to b, anyproperty that either of them have must also be a property of the other. In this case, the term propertymeans exactly what you think it means; e.g. red, even, tall, Hungarian, etc.So, the million dollar question is, by defining =, are our properties now officially axioms? For three of the properties, the answer is no. In fact, because we just defined =, we've turned properties ii), iii), and iv) from above into theorems, not axioms. Why? Because, property iv)still has that = relation in it, which we had to define. So, iv) is a true statement, but we had to use another statement to prove it. That's the definition of a theorem! And, since the qualifier for iv)'s truth was that a ~ b and b ~ a, we can now freely replace b with a in ii), giving us "If a ~ a, then a ~ a." Well, now ii)'s proven as well, but we had to use iv) to do it. Thus, both ii) and iv) are now theorems. Finally, iii) can be proven in a similar was as ii) was, so it, too, is a theorem.However, our definition of = only related a to b, it never related a to itself. Thus, we need to include i), from above, as an axiom.Just for kicks, let's try plan 3) too.The idea here is to make = a part of our signature, which means now we don't need to define it. In fact, we can't define it if we put it in our signature; because by placing it there, we're assuming that it's understood without definition. Therefore, iv) must now be assumed to be true, because we have no means to prove it; that sounds like an axiom to me! However, just like before, we can prove both ii) and iii)through the use of iv), so they get relegated back to the land of theorems and properties. Interestingly though, iv)makes no mention of reflexivity, and since our formal definition of = is gone, we have no way to prove i). Once again, we have to assume that it's true. Thus i) is an axiom as well.So, to paraphrase our two separate situations:In order for the relation ~ to be considered an equality relation between the objects a and b, oneaxiom must be satisfied if we define =:1) For all a, a ~ a,as well as three theorems:1) If a ~ b, then b ~ a2) If a ~ b and b ~ c, then a~ c, where c is another object3) For all a and b, if a ~ b and b ~ a, then a = b.Additionally,In order for the relation ~ to be considered an equality relation between the objects a and b, twoaxioms must be satisfied if we put = into our signature:1) For all a, a ~ a2) For all a and b, if a ~ b and b ~ a, then a = b,as well as two properties:1) If a ~ b, then b ~ a2) If a ~ b and b ~ c, then a~ c,where c is another object.What the one right above did is include "=" into our formal language, but "=" is equality, so he actually came up with a fairly well axiom before he finishes with the circular looking one.His axiom: We say ~ is an equality relation means whenever x ~ y, for any condition P, P(x) iff P(y)The axiom is the bolded part.After discussion with my Math prof. this morning, that axiom becomes a properties follows from this more formal definition. It does not need to include any more things then what we already have for the formal language.We say ~ is an equality relation on a set A if (a set is something that satisfies the set axioms)For any element in A, a ~ a.If follows that P(a) is true and y ~ a, then P(y) is also true, vice versa. Because in this case, y has to be a for it to work.You might argue well the definition for an equivalence relation have this statement in it too, does that mean equivalence IS equality?No! It's the other way around, equality is equivalence. Equality is the most special case for any relation, say *, where a * a.Take an equivalence relation, say isomorphisms for instance (don't know what that word mean? Google or as it on this website), we know any linear transformation T is isomorphic to T, in particular this isomorphism IS equality. Of course it would be boring if isomorphism is JUST equality, so it's MORE.The other axioms in a definition of a relation are to differ THEM from equality, because equality is the most basic. Equality must always be assumed, it always exist, any other relation is built upon it. It is the most powerful relation, because ALL relations have it. (I mean all relations, say *, such that a * a for all a must at least be equality)
Axioms for functional dependencies and inclusion dependencies?
4. Functional dependencyIn relational database theory, a functional dependency is a constraint between two sets of attributes in a relation from a database.Given a relation R, a set of attributes X in R is said to functionally determine another set of attributes Y, also in R, (written X → Y) if, and only if, each X value is associated with precisely one Y value; R is then said to satisfy the functional dependency X → Y. Equivalently, the projection is a function, i.e. Y is a function of X.[1][2] In simple words, if the values for the X attributes are known (say they are x), then the values for the Y attributes corresponding to x can be determined by looking them up in any tuple of Rcontaining x. Customarily X is called the determinant set and Y the dependent set. A functional dependency FD: X → Y is called trivial if Y is a subset of X.The determination of functional dependencies is an important part of designing databases in the relational model, and in database normalization and denormalization. A simple application of functional dependencies is Heath's theorem; it says that a relation R over an attribute set U and satisfying a functional dependency X → Y can be safely split in two relations having the lossless-join decomposition property, namely into where Z = U − XY are the rest of the attributes. (Unions of attribute sets are customarily denoted by mere juxtapositions in database theory.) An important notion in this context is a candidate key, defined as a minimal set of attributes that functionally determine all of the attributes in a relation. The functional dependencies, along with the attribute domains, are selected so as to generate constraints that would exclude as much data inappropriate to the user domain from the system as possible.A notion of logical implication is defined for functional dependencies in the following way: a set of functional dependencies logically implies another set of dependencies , if any relation R satisfying all dependencies from also satisfies all dependencies from ; this is usually written . The notion of logical implication for functional dependencies admits a sound and complete finite axiomatization, known as Armstrong's axioms.Properties and axiomatization of functional dependenciesGiven that X, Y, and Z are sets of attributes in a relation R, one can derive several properties of functional dependencies. Among the most important are the following, usually called Armstrong's axioms:[3]Reflexivity: If Y is a subset of X, then X → YAugmentation: If X → Y, then XZ → YZTransitivity: If X → Y and Y → Z, then X → Z"Reflexivity" can be weakened to just , i.e. it is an actual axiom, where the other two are proper inference rules, more precisely giving rise to the following rules of syntactic consequence:[4].These three rules are a sound and complete axiomatization of functional dependencies. This axiomatization is sometimes described as finite because the number of inference rules is finite,[5] with the caveat that the axiom and rules of inference are all schemata, meaning that the X, Y and Z range over all ground terms (attribute sets).[4]From these rules, we can derive these secondary rules:[3]Union: If X → Y and X → Z, then X → YZDecomposition: If X → YZ, then X → Y and X → ZPseudotransitivity: If X → Y and WY→ Z, then WX → ZThe union and decomposition rules can be combined in a logical equivalence stating that X → YZ, holds iff X → Y and X → Z. This is sometimes called the splitting/combining rule.[6]Another rule that is sometimes handy is:[7]Composition: If X → Y and Z → W, then XZ → YWEquivalent sets of functional dependencies are called covers of each other. Every set of functional dependencies has a canonical cover.
How do you make An integer is divisible by 100 and only if its last two digits are zeros biconditional?
In mathematical logic, An integer A if divisible by 100 iff the last two digits are 0. "iff" stands for "if and only if".
Is the sum of two vectors of equal magnitude equal to the magnitude of either vectors AND their difference root 3 times the magnitude of each vector?
iff the angle between them is 120 degrees
What does IFF mean when used in a biconditional statement?
the statement IFF means "if and only if"
What is the spacing in miles between 7700 emergency decode pulses?
1 Mile -ET1 AIMS MK XII IFF Tech
Universal properties of NAND and NOR gate?
This question really needs a little more context, but an attempt would be: 1: Properties common to both NAND and NOR gates: - they are both electronic logic circuits (as implied by the term "Gate") - they both compute a primitive single valued Boolean function of two or more input terms - they both implement the inverted output version of their primitive (the leading 'N') - there are no families of logic components that implement one and not the other in their catalog 2: differences: - the NAND output is TRUE iff any of its inputs are FALSE - the NOR output is FALSE iff any of its inputs are TRUE - the NAND circuit is much simpler to implement than NOR (NB: the term 'iff' means 'if and only if' - it is not a typo)
The sum of a number and its square is 2?
One -- Well, since we are talking about a second-degree equation, we should at least check if there might be another root (it could be the same). The equation is: x + x^2 = 2 iff x^2 + x - 2 = 0 iff (x+1/2)^2 - 2 - 1/4 = 0 iff (x+1/2)^2 = (3/2)^2 iff x+1/2 = +- 3/2 iff x = -2 or x = 1 Hence, the other solution is -2. Note that -2 + (-2)^2 = -2 + 4 = 2 as required. -- The difference of a number's square and its double is -1? Answer: one (and only one). Check this.
What has the author Roger Iff written?
Roger Iff has written: 'Besteuerung von Versicherungsleistungen und Bankprodukten' -- subject(s): Taxation, Law and legislation, Pensions
What is the syllables for bailiff?
There are 2 syllables. Bail-iff.
What is the purpose of the IFF?
Identification Of friendly and hostile aircraft
Can you be a binweevil mod and how?
No i dont think so only iff you contact binweevils and you would probebly have to be an adult and only iff youv'e never been kicked or banned!!
Is there really some people who are stranded on remote isles?
And how would we know iff they was stranded on remote isles, iff we knew about it, they would not be stranded because we would off rescued them.
How do you get barbiegirls v.i.p. codes?
well iff u dont have your mp3 packet any more u can serch online but iff u have it of a fashion pack there will be a code somewhere on that
