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What else can I help you with?
What is the difference between "if" and "iff" in formal logic?
if is an expression, while iff is a function often with multiple choices.
What is the difference between an outline and prewriting?
prewriting is free writing what you know about a subject, an outline is a formal list with organization and flow.
How do you find a charge of C2?
formal charges. formal charge of each C is 4-2-(6/2)=4-2-3=-1. so the total charge is -2
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)
Which is the opposite and adjacent sides in a triangle?
Each triangle has three sides and three vertices. The opposite side of a triangle is the side that is not adjacent to the specified vertex. The other two sides are adjacent sides to the specified vertex. Circular definition? Yes - Here is the formal definition... Given a triangle with vertices A, B, and C, the side AB is adjacent to the angles ABC and BAC, and it is opposite to the angle ACB.
Difference between formal and informal organisation?
difference between formal organization structure and informal organization structure.
What is the difference between formal and informal rhythm in arts?
formal and informal rhythm
What the balls is the difference between formal and informal?
Formal is polite, informal is rude and obnoxious.
What is the difference between formal and informal slang?
There is no formal slang. Slang is informal language.
Difference between informal and formal proposal?
jdjdjd
What is the difference between formal and informal rules?
pops
What is the difference between formal and contextual criticism?
Nothing.
What is difference between formal and informal barriers to trade?
formal is smart informal isn't smart
What is the difference between it's and it is?
(it is) is more formal than it's
What is the differences between a formal and informal bookkeeping system?
difference between informal business bookkeeping and formal business bookkeeping in there stock
What is the difference between formal dress for a party versus formal dress for debate?
It's the same I think
Difference between formal outline and informal outline?
an formal outline must begin with Roman numeral
