Renegade343 Posted December 25, 2015 Share Posted December 25, 2015 So, someone asked me to solve this: Prove or disprove that for sets B, D, and S∩M, B x (D x (S∩M)) = (B x D) x (S∩M) After a bit, I managed to prove that it is associative (with a small bit of hand-waving because I ran out of space on the paper that I was using to prove it, but the answer should be intuitively clear after an example or two), in which I started to say, "The Cartesian product of sets B, D, S and..." before that person let out a small chuckle. That is when I realised what he wanted me to say, to which I played a counter: Intersection of sets is commutative, hence the set S and M = the set M and S (because at the time, I assumed that when I said "and" between two sets, he would know that I am referring to the intersection of the two sets). Hence, I stopped myself, then restarted, "The Cartesian product of sets B, D, M and S is associative.". His expression was crushed afterwards. Link to comment Share on other sites More sharing options...
(XBOX)Grihaly Posted December 25, 2015 Share Posted December 25, 2015 Can you explain it in English? Link to comment Share on other sites More sharing options...
011100110110000101101101 Posted December 25, 2015 Share Posted December 25, 2015 'Kay? Now, list me the practical applications this can be used for. I would very much like to know. Link to comment Share on other sites More sharing options...
Renegade343 Posted December 25, 2015 Author Share Posted December 25, 2015 'Kay? Now, list me the practical applications this can be used for. I would very much like to know. Understanding the world, and quite a lot of papers that use advanced set theory. In particular, surreal numbers, which I am currently giving it a second try in fully understanding it. Link to comment Share on other sites More sharing options...
(PSN)Pharen Posted December 25, 2015 Share Posted December 25, 2015 Understanding the world, and quite a lot of papers that use advanced set theory. In particular, surreal numbers, which I am currently giving it a second try in fully understanding it. I am not familiar with this symbol: ∩ Link to comment Share on other sites More sharing options...
Renegade343 Posted December 25, 2015 Author Share Posted December 25, 2015 I am not familiar with this symbol: ∩ Intersection of two sets. Basically, for any two sets A and B, the set A∩B contains elements that appear in both A and B (hence why I sometimes call that notation as simply "A and B"). For instance, if A = {1, 2, 3}, and B = {2, 3, 4}, then A∩B = {2, 3}. Link to comment Share on other sites More sharing options...
(PSN)Pharen Posted December 25, 2015 Share Posted December 25, 2015 (edited) Intersection of two sets. Basically, for any two sets A and B, the set A∩B contains elements that appear in both A and B (hence why I sometimes call that notation as simply "A and B"). For instance, if A = {1, 2, 3}, and B = {2, 3, 4}, then A∩B = {2, 3}. Ah, so this is a graph problem. If I still had the program for my virtual CP330CAS, I could give you a visual representation that is does or does not work, unfortunately I lost the CD key... Edited December 25, 2015 by (PS4)Pharen Link to comment Share on other sites More sharing options...
Renegade343 Posted December 25, 2015 Author Share Posted December 25, 2015 Ah, so this is a graph problem. If I still had the program for my virtual CP330CAS, I could give you a visual representation that is does or does not work, unfortunately I lost the CD key... You do not need a graphing program to solve that (in fact, I find that a crutch for the most part, which bores me). I already solved it algebraically. Link to comment Share on other sites More sharing options...
(PSN)Pharen Posted December 25, 2015 Share Posted December 25, 2015 (edited) You do not need a graphing program to solve that (in fact, I find that a crutch for the most part, which bores me). I already solved it algebraically. This is the calculator I used in my degree: It is hardly a crutch. You can get a virtual version for the PC. All you do is input the equation in the field and it can work out any equation. If the equation is not solvable it will point it out. If the equation is solvable it can also plot the graph for it and give a visual representation. In fact it is so powerful that the tutor of my class asked the board of trustees to ban it for exams. However his request was denied because the 330CAS had prior approval due to the fact it is a readily available device and that you need to learn how to use the advanced functions in it. Edited December 25, 2015 by (PS4)Pharen Link to comment Share on other sites More sharing options...
Renegade343 Posted December 25, 2015 Author Share Posted December 25, 2015 This is the calculator I used in my degree: It is hardly a crutch. You can get a virtual version for the PC. All you do is input the equation in the field and it can work out any equation. If the equation is not solvable it will point it out. If the equation is solvable it can also plot the graph for it and give a visual representation. In the context of set theory, calculators are crutches for the most part because set theory is quite similar to logic and proofs , and having a calculator do that hard work for you just shows your skill with using a calculator, not showing how well (or fast) you can think and analyse the question to prove something is true or false. Also, I have found out from studying other people that an over-reliance on calculators to solve problems tends to lead them doubting themselves a lot more often whenever they encounter a question that the calculator cannot solve, to which I will respond with, "If everything is logical, then your answer is correct.", but this point is tangential to what I want to say. Link to comment Share on other sites More sharing options...
(PSN)Pharen Posted December 25, 2015 Share Posted December 25, 2015 In the context of set theory, calculators are crutches for the most part because set theory is quite similar to logic and proofs , and having a calculator do that hard work for you just shows your skill with using a calculator, not showing how well (or fast) you can think and analyse the question to prove something is true or false. Also, I have found out from studying other people that an over-reliance on calculators to solve problems tends to lead them doubting themselves a lot more often whenever they encounter a question that the calculator cannot solve, to which I will respond with, "If everything is logical, then your answer is correct.", but this point is tangential to what I want to say. But what you don't understand is that this calculator doesn't just come to a conclusion. It also has the power to show all the workings involved. Theory is fine if you have hours to waste on a single problem but in the real world a device like this is the obvious choice in a business environment. Link to comment Share on other sites More sharing options...
Ampoth Posted December 25, 2015 Share Posted December 25, 2015 (edited) At first glance it seems like the Cartesian product might be associative, but since A x B := {(a,b): where a in A, and b in B}, you have through the use of the definition that ((b,d),sm) != (b,(d,sm)) assuming that A, B, and S∩M are all nontrivial. This simple counter example is enough to disprove the conjecture. Also, Pharen I can't see how a calculator would be useful at all for this, I'm not sure you understand the problem. Edited December 25, 2015 by Ampoth Link to comment Share on other sites More sharing options...
Freelancer27 Posted December 25, 2015 Share Posted December 25, 2015 Hence, I stopped myself, then restarted, "The Cartesian product of sets B, D, M and S is associative.". His expression was crushed afterwards. The simplest words from a correct answer is really worth saying for. Prove or disprove that for sets B, D, and S∩M, B x (D x (S∩M)) = (B x D) x (S∩M) Notably, B x (D x (S∩M)) becomes B x D x S∩M. As the Parentheses only gets the lime-light when B and D gets a similar interaction like S∩M. So, yeah. That easy to find out. Link to comment Share on other sites More sharing options...
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