>Climate change >Resource exhaustion >Microplastics
It seems grim desu, and I worry about this sort of thing a lot. Is there any hope, or are we doomed?
>The life-cycle of these ships is another aspect of the industry that seems to be out of sync with the environment and social justice. What exactly does that look like?
Yep, social justice and environmentalism is hauling us back to 18th century tech levels.
>It would be helpful if you could show us an example of this cube root of 2,
says Kate, one of your eighth grade math students.
>Well, consider a cube with a volume of 2 cubic units. Its side length is the cube root of 2.
>Isn't it circular reasoning to prove the number exists that way?
asks David. >We don't know if a cube with that volume is possible until we've found a cube root of 2.
>Let's try another approach. Make a graph of , then draw a line at . The point where the line meets the curve has x coordinate .
>Why is it okay to just plot a few points and then just guess where the curve goes?
asks Kate. >You would never accept this sort of reasoning in a geometry proof, but when graphing it's apparently okay.
David raises his hand. >I was wondering how the graphing calculator does it, so I asked my uncle who programs computers for a living. He said that it just plots a large number of points and connects consecutive points with the closest thing to a line it can draw. That's not really a smooth curve, it just looks like one.
Can you show a better example of ? What about other numbers David and Kate will learn about, like ln(2), or erf(1), or the sum of 1/2 to the power of every integer which is the ASCII encoding of a valid halting Brainfuck program? How far can you go?
>two sets have the same size if and only if there exists a bijection between them >two sets have the same size if and only if all injective mappings are surjective
Both of these definitions are valid for finite sets. In order to use the second definition, you simply need to show the existence of a bijection, and it can be proven that a bijection implies that all injective mappings are surjective. So is it the first “definition” that is implied from the second, or the second that is implied from the first? Why is one more valid than the other?
Infinite sets rely on using the first definition while ignoring the second. For this reason, infinite sets can appear to have the same size while also having a different size. Example:
Set A = {1, 2, 3, …}
Set B = {0, 1, 2, …}
The mapping x~x-1 is injective and surjective, but the mapping x~x is injective and not surjective. Intuitively, if every element in A maps to an element in B, but 0 still isn’t mapped to, set B is bigger than set A. This contradicts the mapping x~x, for which it seems that the sets have the same size.
According to definition 1, we simply ignore the contradiction and declare them equal. That is the cause of paradoxes related to infinity. Why do we not admit there is a contradiction and say that it makes no sense to talk about the cardinality of an infinite set? Or that infinite sets themselves are an unfounded axiom?
Trying to decide on a major. Engineering/Stats or Premed. I know I have to choose one or the other as Engineering and Stats are notoriously tricky, and it would be a stretch to have the high GPA required for Med School. So I haven't looked at other majors. And if I don't get through pre-med, I don't want to be stuck—such a tough decision.
EDIT: I know premed is not a major. I just meant that everyone I have spoken to says if you want to apply to med school, pick an easier major than Engineering or Stats. So people who think they will use to medical school, what is your major? What will you do if you don't get into Medical School? And people who chose an easier major with the plan of going to Medical School and didn't apply/get accepted are disappointed that you changed their major for Medical School.
