The proof for P(n): "Every set w/ n ele has 2^n subsets" was quite elegant.
- Since the end of CSC165, I have had the trouble of expressing my intuition in mathematical terms, and the bijective fn f:s^- -> s^+, f(s) = s U x (mapping from the set that doesn't include x (ele n+1) to the set that does include x (ele n+1)) was precisely (#-1) the fn needed to do so in this proof. The book itself didn't formalise the function though. I think the fn itself takes some time to create, so hopefully a more informal (i.e. ..."worded") proof of the bijection between the two sets will suffice. I practiced creating such a fn for the first assignment - hopefully it will do.
- Let S an arbitrary set with n ele. I do not know exactly how to express in mathematical symbolism: the set of subsets of S that don't contain the ele x and the set of subsets that do contain the ele x. Perhaps the former will be {/forall s /in Powerset(S): x /not/in s} and the latter will be {/forall s /in Powerset(S): x /in s}. (#-2)
The proof for P(n): "12^n -1 is a multiple of 11".
- Really a classic example of mathematical creativity. I really have to take note of this, since you can't really work forwards or backwards.
The proof for P(n): "Units digit of 4^n \in {1, 4, 6}".
- A simple proof, but when I first tackled it I couldn't express my intuition in mathematical terms.
- In tutorial, the TA expressed it as \exists z \in Z.\exists t \in {1, 4, 6}. 4^n = 10z + t (#1). I like that (#2).
Proof of A1Q4.
- I just liked the fact that we practiced claim strengthening. It really was interesting. I wonder if something will come up where we prove one strong claim version, then we need to use a variety logical rules to prove from that strong claim the weakened claim. Like if we needed to use deMorgan's and contrapositive.
Proof of A1Q3.
- So awesome. Really showed me that I need to organise (and write neatly...) when I play around with a claim.
#0: I need practice on structural induction and well-ordering. I'll look at the textbook later...
#1: Professor Ellen used '.' instead of ',' in between quantifies and predicates. I like this because many I suck at writing ',' and there are claims that involve sets of integers as this one, so '.' allows me to focus more on the claim.
#2: Dear TA, let's please proceed faster in tutorials - as much as I do enjoy how carefully you overview proofs.
#-1: Since I was brainwashed with Physics in high school, I still believe the term accurately should be used in place of precisely, as bad as it sounds. On the other hand, no one can ever know if an answer is close to "truth" or not.
#-2: This reminds me, there is so much power into solving problems by playing around with them by hand (#-3), i.e. in assignment 1 question 3, writing down the lists of 2-subsets and 3-subsets (almost like art) can really open up your mind. I have to explore this more myself.
#-3: I was going to refrain from falling into using boldface, strikethroughs, etc. since not using them really does help me gain wisdom in choosing appropriate words. More power to technology.
The proof for P(n): "12^n -1 is a multiple of 11".
- Really a classic example of mathematical creativity. I really have to take note of this, since you can't really work forwards or backwards.
The proof for P(n): "Units digit of 4^n \in {1, 4, 6}".
- A simple proof, but when I first tackled it I couldn't express my intuition in mathematical terms.
- In tutorial, the TA expressed it as \exists z \in Z.\exists t \in {1, 4, 6}. 4^n = 10z + t (#1). I like that (#2).
Proof of A1Q4.
- I just liked the fact that we practiced claim strengthening. It really was interesting. I wonder if something will come up where we prove one strong claim version, then we need to use a variety logical rules to prove from that strong claim the weakened claim. Like if we needed to use deMorgan's and contrapositive.
Proof of A1Q3.
- So awesome. Really showed me that I need to organise (and write neatly...) when I play around with a claim.
#0: I need practice on structural induction and well-ordering. I'll look at the textbook later...
#1: Professor Ellen used '.' instead of ',' in between quantifies and predicates. I like this because many I suck at writing ',' and there are claims that involve sets of integers as this one, so '.' allows me to focus more on the claim.
#2: Dear TA, let's please proceed faster in tutorials - as much as I do enjoy how carefully you overview proofs.
#-1: Since I was brainwashed with Physics in high school, I still believe the term accurately should be used in place of precisely, as bad as it sounds. On the other hand, no one can ever know if an answer is close to "truth" or not.
#-2: This reminds me, there is so much power into solving problems by playing around with them by hand (#-3), i.e. in assignment 1 question 3, writing down the lists of 2-subsets and 3-subsets (almost like art) can really open up your mind. I have to explore this more myself.
#-3: I was going to refrain from falling into using boldface, strikethroughs, etc. since not using them really does help me gain wisdom in choosing appropriate words. More power to technology.
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