What is the difference between the transitive property (if a=b and b=c then a=c) and substitution (if a=b then a can be replaced by b in any equasion and vice versa), and why can't I use the two interchangably in proofs involving parallelograms?
If I cannot use the two interchangably, how do I know when to use the appropriate proof?
I apologize, but I don't quite understand your answer. I have an assignment in school that requires that I complete some proofs involving parallelograms, and I used the substitution method to complete it. Here is the problem, and I want to know why I cannot say "substitution" instead of "transitive"
--F_______________E______D
--/-----------------/-------/
-/-----------------/-------/
/_______________/______/
A----------------B------C
To prove: ACDF is a parallelogram
ABEF is a parallelogram; EB is parallel to FE--------Given
AF is parallel to BE; AB is parallel to FE------------Opposite sides of
-------------------------------------------------parallelogram are parallel
CD is parallel to AF-------------------------------Transitive (why can't it
-------------------------------------------------be substitution instead
-------------------------------------------------of transitive?)
FD is parallel to AC-------------------------------Part of FE and AB
ACDF is a parallelogram---------------------------Definition of Parallelogram
You ask a very important question and you have gotten a correct answer.
I would like to expand on the answer. The answer is at to very heart of the foundations of mathematics. Equality is the idea of things being identical whereas transitivity is a property of relations. Now the identity relation has the transitive property. Having the transitive property allows for “substitutions”.
Here is a simple example. At one time it was common to read that ‘the base angles of an isosceles are equal’. Well of course that is absurd: they are two different angles! It is true that the angles have the same measure and are said to be congruent. Transitivity is a property of the congruence relation. So we can use substitution when dealing with congruent angles.
I am not exactly sure what you are asking thus I will as much as a can even if you were not asking that.
The word rigorous is a English word that mathematicians love to use. Another word mathematicians love to use is intuitive. For example, many many concepts from Calculus, are intuitively understood. Concepts such as: area, volume, length, tangent lines. And for the most part physicists/engineers only need the intuiteve concepts, because they are based on the physical applications of the subject matter. Mathematicians, however, are strange. They cannot, at all, never, ever, woe on their soul, accept anything intuitive. Meaning, if f(x)>g(x) on interval [a,b] then INT(a,b) f(x) > INT (a,b) g(x). The intuition says that since integration is area it makes sense to says that this statement is true. However, a mathemation does not accept such things, he will spend hours and days proving even the most simple of concepts. The reason why is that mathemations do not make proofs based on their intuition (they can use it to construct ideas but not proofs). And in fact, those who study math at a serious level will agree that intution is a dangerous thing. For example, the thing I said about f(x) >g(x) is not acutally true when f and g are discontinous at too many points! Now there is no way human intutition can see something like that. And the beauty is when you prove something rigorously then there is absolutely no way that you are wrong. Another example, everybody uses Euclidean geometry, everybody knows it nobody denys it. However, mathemations found it hard to accept because Euclid does not define rigorously the meaning of a point, a line, a plane,..... And about 2000 years later a mathemation comes along, Hilbert, and he turns it into a rigorous acceptable subject, so rigorous that you do not even need diagrams/pictures to make geometric proofs! However, his geometry is extremely complicated and is at all not necessary for people to learn if they just want to solve physical problems. Mathemations have a thing in them that they need to see a rigorous developement of the subject matter. That is what makes them so different from every single disciple that exists. Now that is what I mean by rigorous. The transitive property is a relation, something that relates two elements/objects to each other. Thus we say if a related to b and b related to c then a related to c. That is a rigorous statement. However, the substitution property states you can replace similar things into each other. The difficutly is what does it mean "similar"? Mathemations therefore disfavor it. One last example, when Calculus was first introduced by Newton and Leibniz mathemations had difficutly accepting it because certain concepts were never defined. Like what does it mean a limit "approaches", what does "approaches" mean. Only later on did mathemations improve Calculus so it is acceptable among them. Many people (non-math) think proving these simple looking things is stupid but I disagree not only did they make Calculus beautiful they discovered several shattering statements.
I don't understand a thing about what you just said. Or maybe I do. So you and your kind think differently from us. So you and I do not speak in the same frequency. I am intuitive, and you are rigorous, whatever that means. So when you said I did not say when a=b, then my sollution on one interation question is wrong. And I said you were wrong in saying that. So you knew I will not accept what you remarked because we do not "speak/thgink" the same, then why did you say that?
I now know we do not think the same, so whatever you comment on my solutions means _______ to me more. So expect to receive ______ more responses from me if you do not get it yet.
That group of people are called, mathematicians.
Yes, if you want to be a mathematician, then you must be rigorous (almost always).So you and I do not speak in the same frequency. I am intuitive, and you are rigorous, whatever that means.
Because people need to use theorems in the way they are proved.So you knew I will not accept what you remarked because we do not "speak/thgink" the same, then why did you say that?
For example, the fundamental theorem says:
INT f(x) dx = F(b) - F(a).
But that is only true for continous functions, it if is not it might fails to work. Thus, even if someone solved a problem via this equation and gets the right answer does not mean his appraoch was sound , justified by the theorem.
All the top mathemations (modern) follow the same standard approach, they are not interested in applications.
Furthermore, when mathemations discuss things among themselves it is always a complete abstraction.
Buy yourself a book on a higher math subject, say "Analysis" and perhaps you will like it (cheap book, and well written, hopefully you have time).
No! But they are very well familar with the pure side as well. I was not saying there is something wrong with applying, I was saying that everyone who does math today is familar with the abstract concepts, they decipher them into applied problems. But! You do not have a quantum field physicist who is considered a mathemation.
I can analyze very well, even Math propblems. I mean useful Math.
About your kind of Math? No. It is, uh, well,.........(never mind.)
[I was tempted to say "useless" but I stopped myself. There are bad words that might not appear so, but they are. I won't be caught saying those.]
They are all useful.
Examples,
1)Einstein found a way of describing space through "Differencial Geometry", a math topic which probably had no application until then.
2)In Quantum Mechanics, Homotopy groups appear, another supprising application of pure math.
3)Mathematical Logic can sometimes be applied to computer science.
...
Now, if all these people had your approach and said, well, let me never use it. That will never had happened.
As I said, the is a standard math, all mathematicians learn.About your kind of Math?
You should first learn these different fields of math first before you can complain about them and say they are useless.No. It is, uh, well,.........(never mind.)