What did I find difficult?
I originally had difficult understanding existence proofs. The reason for this? The extent of my proving abilities consisted mainly of contradiction, direct proofs, contrapositive. You could say that learning existence proofs was like doing a new exercise - it was weird working through them and understanding it, but once I got it, I liked it. After reading, I think existence proofs are pretty self explanatory. My understanding of them is that I just have to prove one case - that such a case exists. I like the story about the mathematician David Hilbert:
"There is at least one student in this class... let us name him 'X'... for whom the following statement is true: No other student in the class has more hairs on his head than X. Which student is it? That we shall never know; but of his existence we can be absolutely certain."
Before reading the example, I was strictly going off of the definition of existence proofs. But this example helps me understand that we just need to prove the existence of something. I really like it.
What did I find interesting?
Disproving existence statements. I like the existence way of going about, solving a proof. I then find it fascinating that you can disprove these existence statement. I am fascinated with the way set notation works, and how the negation of an existential quantifier turns that quantifier into a universal quantifier. I really like this way of solving a problem, and proving whether something exists or whether it doesn't exist. I like that I am able to apply what I learned earlier on in the course and use it in such a way that it makes me understand what is going on.
I originally had difficult understanding existence proofs. The reason for this? The extent of my proving abilities consisted mainly of contradiction, direct proofs, contrapositive. You could say that learning existence proofs was like doing a new exercise - it was weird working through them and understanding it, but once I got it, I liked it. After reading, I think existence proofs are pretty self explanatory. My understanding of them is that I just have to prove one case - that such a case exists. I like the story about the mathematician David Hilbert:
"There is at least one student in this class... let us name him 'X'... for whom the following statement is true: No other student in the class has more hairs on his head than X. Which student is it? That we shall never know; but of his existence we can be absolutely certain."
Before reading the example, I was strictly going off of the definition of existence proofs. But this example helps me understand that we just need to prove the existence of something. I really like it.
What did I find interesting?
Disproving existence statements. I like the existence way of going about, solving a proof. I then find it fascinating that you can disprove these existence statement. I am fascinated with the way set notation works, and how the negation of an existential quantifier turns that quantifier into a universal quantifier. I really like this way of solving a problem, and proving whether something exists or whether it doesn't exist. I like that I am able to apply what I learned earlier on in the course and use it in such a way that it makes me understand what is going on.
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