The previous week we learned about structuring proofs, and this week we learned strategies to doing actual proofs. Proof structures aren't too complicated to learn, but it was good practice to do several of them because it was very easy for me at first to just skip some necessary steps in writing a proper proof. We also learned that you can use the contrapositive to prove the statement itself, which seems like a good strategy in situations where the statement feels more ambiguous or difficult to prove. That is, if the statement is P -> Q, then the contrapositive is not(Q) -> not(P).
If you have a universal statement such as the one used in class:
"There are 5 boxes that have a total of 51 balls. Prove that there is a box with at least 11 balls in it"
So the contrapositive to this would be something like :
"There does not exist a box with at least 11 balls, therefore the 5 boxes do not have a total of 51 balls."
or in other words:
"For 5 boxes, if all boxes have less than 11 balls, the sum of all balls cannot be 51."
From my understanding, the reason for proving the contrapositive would be that to prove the statement you would actually have to directly show that a box has at least 11 balls in it, so you would need to know the set of all balls in each box (I think?).
So, given not(Q), or that each box can have at most 10 balls, that means the highest possible sum for 5 boxes with at most 10 balls each is 50. Therefore, in order to have 51 balls, one box has to have at least 11 balls in it.
Now, moving on to actual proofs, some of this was simple and some a lot more tricky. The concept of proving a statement using the definition function made a lot of sense. It seems like if a statement is being made using a single function, knowing the definition of the function alone can be all you need to prove or disprove that statement (as I said, if the statement is just about that single function, but I wont believe this for certain until I know it to be true for sure). At least, when a function definition is all you have, the strategy of manipulating that function into different forms is the best way to prove/disprove.
Disproving was also somewhat straightforward conceptually. You're simply proving the negation. I think doing so is much easier when disproving a universal statement, because it's much simpler to say that something exists than that everything exists. That being said, I think the above strategy of proving the contrapositive when faced with a difficult statement to prove can also be used when proving a negation. That is, to prove the contrapositive of a negation.
So, lets say my statement is:
There exists a penguin that is a mammal. P -> Q
the negation of this would be : P -> not(Q)
All penguins are not mammals.
the contrapositive of the negation would be : Q -> not(P)
If it is a mammal, then it isn't a penguin.
Probably not the best example, but I think the concept of using the contrapositive still applies.
So, the rest of the class had a lot of math and stuff about limits. It has been a while since I've done calculus, but it shouldn't be too tricky to remember some things to understand this stuff better. For now though, it was a little bit daunting.
If you have a universal statement such as the one used in class:
"There are 5 boxes that have a total of 51 balls. Prove that there is a box with at least 11 balls in it"
So the contrapositive to this would be something like :
"There does not exist a box with at least 11 balls, therefore the 5 boxes do not have a total of 51 balls."
or in other words:
"For 5 boxes, if all boxes have less than 11 balls, the sum of all balls cannot be 51."
From my understanding, the reason for proving the contrapositive would be that to prove the statement you would actually have to directly show that a box has at least 11 balls in it, so you would need to know the set of all balls in each box (I think?).
So, given not(Q), or that each box can have at most 10 balls, that means the highest possible sum for 5 boxes with at most 10 balls each is 50. Therefore, in order to have 51 balls, one box has to have at least 11 balls in it.
Now, moving on to actual proofs, some of this was simple and some a lot more tricky. The concept of proving a statement using the definition function made a lot of sense. It seems like if a statement is being made using a single function, knowing the definition of the function alone can be all you need to prove or disprove that statement (as I said, if the statement is just about that single function, but I wont believe this for certain until I know it to be true for sure). At least, when a function definition is all you have, the strategy of manipulating that function into different forms is the best way to prove/disprove.
Disproving was also somewhat straightforward conceptually. You're simply proving the negation. I think doing so is much easier when disproving a universal statement, because it's much simpler to say that something exists than that everything exists. That being said, I think the above strategy of proving the contrapositive when faced with a difficult statement to prove can also be used when proving a negation. That is, to prove the contrapositive of a negation.
So, lets say my statement is:
There exists a penguin that is a mammal. P -> Q
the negation of this would be : P -> not(Q)
All penguins are not mammals.
the contrapositive of the negation would be : Q -> not(P)
If it is a mammal, then it isn't a penguin.
Probably not the best example, but I think the concept of using the contrapositive still applies.
So, the rest of the class had a lot of math and stuff about limits. It has been a while since I've done calculus, but it shouldn't be too tricky to remember some things to understand this stuff better. For now though, it was a little bit daunting.
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