So now that the assignments are all over and submitted, I thought I'd dedicate this week's course logs to stuff about the assignment, to things that were challenging, confusing, or just plain easy.
To start off with question one, it didn't feel too challenging. The way I solved the problem was to first write out each of the statements with mathematical symbols, and then negate the statements. After that, I translated the negation into English. While it would have been easier to just figure out the negation in english first and then turn that into a symbolic representation, I felt that it would be good practice of using symbols to do it the other way around, and it would also help avoid misinterpreting some of the vague aspects of English.
So, for example, problem d:
There are diagonal acronyms that are bifurcated.
\thereexists a \element of A| D(a) ^ B(a)
So, to negate that, I said that
d. All diagonal acronyms are not bifurcated.
∀a\elementof A| D(a) ^ ¬B(a)
Moving on to question 2, I'd have to say I enjoyed this part most of all, even though part d was challenging and a bit tricky. So, I had to decide whether the statements were equivalent to the converse, contrapositive, and negation. The first 3 seemed straightforward, but the last one was actually an inversion. My initial reaction was to think that the question was written incorrectly, but I realized that it asked for "equivalence", which I believe refers to the truth value in relation to the original statement. An inversion is just a converse of the contrapositive, and since the contrapositive depends on the truth value of the original statement, it's converse, should function in the same way as the converse of the original statement. So, I determined that the inverse is equivalent to the converse.
I don't really have anything to say about question 3, and the only thing about question 4 that bugged me was drawing venn diagrams in a word/text processor. As for 5 though, this part was tricky. There's still plenty of things that I have to go over, but my general logic was that for one to imply the other, it has to be its subset, so I just went with answering it from there. The only thing I couldn't figure out at all was the final one, the empty set. I hope that this gets answered for me eventually though!
To start off with question one, it didn't feel too challenging. The way I solved the problem was to first write out each of the statements with mathematical symbols, and then negate the statements. After that, I translated the negation into English. While it would have been easier to just figure out the negation in english first and then turn that into a symbolic representation, I felt that it would be good practice of using symbols to do it the other way around, and it would also help avoid misinterpreting some of the vague aspects of English.
So, for example, problem d:
There are diagonal acronyms that are bifurcated.
\thereexists a \element of A| D(a) ^ B(a)
So, to negate that, I said that
d. All diagonal acronyms are not bifurcated.
∀a\elementof A| D(a) ^ ¬B(a)
Moving on to question 2, I'd have to say I enjoyed this part most of all, even though part d was challenging and a bit tricky. So, I had to decide whether the statements were equivalent to the converse, contrapositive, and negation. The first 3 seemed straightforward, but the last one was actually an inversion. My initial reaction was to think that the question was written incorrectly, but I realized that it asked for "equivalence", which I believe refers to the truth value in relation to the original statement. An inversion is just a converse of the contrapositive, and since the contrapositive depends on the truth value of the original statement, it's converse, should function in the same way as the converse of the original statement. So, I determined that the inverse is equivalent to the converse.
I don't really have anything to say about question 3, and the only thing about question 4 that bugged me was drawing venn diagrams in a word/text processor. As for 5 though, this part was tricky. There's still plenty of things that I have to go over, but my general logic was that for one to imply the other, it has to be its subset, so I just went with answering it from there. The only thing I couldn't figure out at all was the final one, the empty set. I hope that this gets answered for me eventually though!
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