This week the material has definitely been more challenging than the previous two weeks. I am still wrapping my head around how to implement a lot of the symbols correctly, so it has been difficult to fully understand lecture this week. I did not get the quiz question correct, but I learned a lot in the tutorial and from my mistakes, so I am not very concerned about it. My answers to the questions in the exercise before the tutorial were not very well developed, but redoing the questions after my answers were much better syntactically and semantically. I was previously falling victim to doing things like x>1 when referring to a prerequisite, thinking that it meant there was more than one prerequisite when in fact it means nothing at all.
By taking mathematical expression at the same time as calculus and linear algebra, I am seeing a lot of overlap in the principles of mathematical writing. However, mathematical expression has taken this subject much further than the other two courses mentioned which gives me a great background when writing in symbols for them. Another benefit is mathematical expression gets you used to interpreting math notation which is very common in my linear algebra and calculus textbooks.
To practice this weekend I am going to work through proving that the arithmetic laws that can be transferred to logic are in fact true. I found it very useful when Professor Heap was using Venn diagrams in his explanations, so I will try to devise the proofs in the same way. As well I intend to start the assignment.
Friday, 26 September 2014
Friday, 19 September 2014
Logic Problems
On the first weekend of this semester I enjoyed going through the logic problems in chapter 1 of the course notes, although some of them I found quite frustrating. The first problem that I came across was the infamous Monty Hall problem renamed 3 Boxes in the course notes. It is my understanding that when this problem was first presented even PhD's in mathematics could not agree on a solution. My reasoning behind the solution is that if you had a higher chance of selecting an empty door on your first choice, then it is more likely the game show host has shown you the location of the other empty door, thus giving you free probability. Therefore you should switch doors. It certainly isn't an entirely new problem with new parameters when the host shows you the empty door as some people have suggested, and running computer simulations confirm this. The next problem that I chose to tackle was the 2 Labelled Boxes Surprise.
Solution:
The problem provides the reader with two explanations that appear to be foolproof yet don't yield the correct solution. They almost form a mental block when you are trying to reason your way through the problem on your own because you are constantly thinking back to them. The first step you must take in solving this problem is realizing that the box labelled "the prize is not here" must be true in order to have not been lied to. From here I reverse engineered a solution that will mean the boxes have not lied to you. The solution that I came up with is that the box that says "exactly one statement on the boxes is false" is in fact a false statement because both statements are true. This box being true because the statement itself is false, it is circular reasoning, but it is what I came up with. This solution is before I have learned all the techniques taught in 165 so it may not be a perfect explanation or even correct. Does anyone else have a better explanation?
The course work up to the end of the second week hasn't been very difficult, for me it is mostly just a matter of learning all of the terminology. This weekend I am going to make a conscious effort to review notation because it has been a while since I have taken a formal math course. Having written the GMAT and scoring very well I am used to logic problems to some degree. Although most of the logic problems on the GMAT just related to whether a solution could be obtained with statements presented and not what the actual answers were. So this course will take my logic abilities further.
Solution:
The problem provides the reader with two explanations that appear to be foolproof yet don't yield the correct solution. They almost form a mental block when you are trying to reason your way through the problem on your own because you are constantly thinking back to them. The first step you must take in solving this problem is realizing that the box labelled "the prize is not here" must be true in order to have not been lied to. From here I reverse engineered a solution that will mean the boxes have not lied to you. The solution that I came up with is that the box that says "exactly one statement on the boxes is false" is in fact a false statement because both statements are true. This box being true because the statement itself is false, it is circular reasoning, but it is what I came up with. This solution is before I have learned all the techniques taught in 165 so it may not be a perfect explanation or even correct. Does anyone else have a better explanation?
The course work up to the end of the second week hasn't been very difficult, for me it is mostly just a matter of learning all of the terminology. This weekend I am going to make a conscious effort to review notation because it has been a while since I have taken a formal math course. Having written the GMAT and scoring very well I am used to logic problems to some degree. Although most of the logic problems on the GMAT just related to whether a solution could be obtained with statements presented and not what the actual answers were. So this course will take my logic abilities further.
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