One of the challenging parts of the proof unit so far has been that much of the technique seems jarringly obvious when it is explained, yet in practice it seems like it would be quite obscure and very easy to fail to see. Obviously, I will have to explain this thought a bit...
In a recent lecture, Danny went over several "proof techniques" including direct proof, indirect proof, inference rules, existential instantiation, disjunction elimination, implication elimination, universal elimination, etc. These, the way he put it, are very obvious. Universal elimination means that if you know every element of a set has a certain property, then you know that a certain given element of that set also has that property. This is the kind of thing that seems to go without saying. Indirect proof is the practice of assuming the negation of the consequent and deriving the negation of the antecedent. These concepts, as well as many others that we have learned, seem very obvious when simply written out.
The problem comes with the application of these concepts. So far, with the proofs I have encountered, I basically go right for the direct proof and hope it works. If I can't get it to work on the first try, I go back to the "scratch work" and then hope to try again. At this point, I lack the confidence to know that certain situations calls for other proof techniques. It is almost as if they seem so obvious and intuitive to me that I do not know what really sets them apart from each other, and how they are appropriate for certain unique circumstances and not for others. I think I have only done one or two indirect proofs in class/tutorial, and when I did, it was because I had been told by the instructor/TA that indirect proofs were necessary for the given question. In a test or exam, I am not so sure that I would be able to clearly tell that indirect proof or any other sort of technique that is not a simple direct proof is necessary. I also would have a hard time discerning between all of these techniques and knowing exactly which one I was using, and I do not know whether or not that is entirely necessary or not.
Monday 27 October 2014
Saturday 18 October 2014
SLOG week 6
This week we discussed the concept and definition of a limit. I like limits. The discussion of them has reminded me of what we discussed so long ago in the first week of class, namely, absolute precision.
Visualizing and "feeling" a limit is quite simple for a given function. If the limit of f is, say, 20 as x approaches 10, you can easily imagine the line representing the function getting closer and closer and closer and closer to 20 as you start just to the left of 10 and move along the line. It is intuitive, like many concepts in maths.
The definition of a limit, on the other hand, is not very intuitive at first glance. That is why it makes me think of the absolute certainty of it. It looks unnatural when you look at it the first few times, yet it makes so much sense and it successfully lays out the concept with such precision. If you were to say "well, the function just approaches the limit as x gets closer to the given number", then there are many ways in which people might misunderstand you. Such is the imprecision of the English language; you can explain mathematical concepts, but not without ambiguity. The precision of the mathematical symbols and definitions, however, means that,as long as the symbols and equations are understood, the concept will be captured with complete accuracy. It is not ambiguous.
Visualizing and "feeling" a limit is quite simple for a given function. If the limit of f is, say, 20 as x approaches 10, you can easily imagine the line representing the function getting closer and closer and closer and closer to 20 as you start just to the left of 10 and move along the line. It is intuitive, like many concepts in maths.
The definition of a limit, on the other hand, is not very intuitive at first glance. That is why it makes me think of the absolute certainty of it. It looks unnatural when you look at it the first few times, yet it makes so much sense and it successfully lays out the concept with such precision. If you were to say "well, the function just approaches the limit as x gets closer to the given number", then there are many ways in which people might misunderstand you. Such is the imprecision of the English language; you can explain mathematical concepts, but not without ambiguity. The precision of the mathematical symbols and definitions, however, means that,as long as the symbols and equations are understood, the concept will be captured with complete accuracy. It is not ambiguous.
Saturday 11 October 2014
SLOG week 5
I do not want to spend all of my SLOG posts praising the course and the teaching, but for now, I will, because I am still enjoying it and learning well. So far, my favourite aspect of the course has been the way that it has focused--just as the course title implies--on reasoning and logic more than facts and memory. I have learned, so far, to think a certain way, and I find that the course material has overseen this exceptionally well. When I began the first assignment, I remembered much of what I had learned in class, but I had not used it yet. The assignment forced me to go back, review, apply the concepts, and, finally, to internalize them. I was immersed in the practice of Boolean algebra and implication, and I really had to understand them to complete the assignment. Because of the assignment's forcing me to internalize the course concepts, it prepared me excellently for the test.
My thoughts on the first test: I felt confident and comfortable with the material. After completing the first assignment, I felt totally prepared. The only real problem I had with it was that reading the Python functions and trying to follow them while under time constraints was a bit difficult, especially because I had to remember which one fit with which definition and then I had to retain that information to apply it to the next question. I basically ended up doing this by sorting it out first, then writing it down, and then quickly referencing my writing on the next page to recall the necessary function. This turned out to be a bit confusing, and I think I might have lost a mark just because I was flipping between pages and referencing the things I had scrawled next to each function without necessarily thinking it through each time. Either way, though, it was an interesting test.
The last question was strange when I first read it. Presumably just as it was meant to, it pushed my logical intuition at first, because it called for sets P, Q, and D such that one of the following is true:
My thoughts on the first test: I felt confident and comfortable with the material. After completing the first assignment, I felt totally prepared. The only real problem I had with it was that reading the Python functions and trying to follow them while under time constraints was a bit difficult, especially because I had to remember which one fit with which definition and then I had to retain that information to apply it to the next question. I basically ended up doing this by sorting it out first, then writing it down, and then quickly referencing my writing on the next page to recall the necessary function. This turned out to be a bit confusing, and I think I might have lost a mark just because I was flipping between pages and referencing the things I had scrawled next to each function without necessarily thinking it through each time. Either way, though, it was an interesting test.
The last question was strange when I first read it. Presumably just as it was meant to, it pushed my logical intuition at first, because it called for sets P, Q, and D such that one of the following is true:
- There exists an x in D such that P if and only if Q.
- There exists an x in D such that P and Q.
I understood the difference, but at first, I could not quite think of an example of sets that would fit. Eventually, I realized that picturing it as a Venn Diagram was really easy: if D was the universe, in the first one, the only occupied part of the diagram would be the overlap between the circles. On the second, the circles should overlap, but some other part of the diagram would have to be occupied as well. This led me to the easiest thing I could think of, which is a solution that would make P and Q true, but such that P is not necessarily dependant on Q and vice versa. This satisfied the question, though I cannot remember exactly what I chose when writing.
The other way to satisfy the question, of course, would be to make the first statement true and the second false. The only way I can think of to do this is to make it true through vacuous truth: if every element in D falls under neither P nor Q, then the statement will be true and the second statement will be false.
Taking a look at some other SLOGS, I see that there generally, people were comfortable with Test 1. http://slog-christone.blogspot.ca/ here is one (look under week 5). This seems inevitable; though I congratulate myself for finding it easy, the fact is that it was easy because it was very much representative of what we have all learned so far, and no more than that. It did not really "stretch" our understanding of the concepts, so whoever understands the course well at this point was bound to do well.
Taking a look at some other SLOGS, I see that there generally, people were comfortable with Test 1. http://slog-christone.blogspot.ca/ here is one (look under week 5). This seems inevitable; though I congratulate myself for finding it easy, the fact is that it was easy because it was very much representative of what we have all learned so far, and no more than that. It did not really "stretch" our understanding of the concepts, so whoever understands the course well at this point was bound to do well.
Saturday 4 October 2014
SLOG week 4
I am enjoying proofs. Last year I took MAT137, and, though that was a real maths course, I think this course is introducing me to the concept better than that one did. Of course, we have only done very, very simple proofs so far, all of which depend on algebra that would come easily to most people, like showing that if y = 10x, then y(squared) = 100x(squared) or something like that.
Still, though, I admire the elegance of the process: you assume some antecedent, and if you can naturally derive the consequent from that antecedent, you can prove the implication. The few that we have tried in class have been simple enough and very understandable. I look forward to getting more advanced.
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