Hey guys! So, this time, we’re going look at other methods we can use to construct proofs when just deriving from the premises isn’t enough.
Conditional Proof (CP)
Basically, you use this method when the conclusion or a part of the conclusion you want is a conditional. This makes it so you assume the predicate in order to derive the consequent. Here’s an example:
Indirect Proof (IP)
For this method, you use this primarily when the conclusion is a negated statement. You assume the un-negated form of the conclusion and attempt to find a contradiction so that the assumption is false, thus ending at the negated form. It also works the other way around where the conclusion isn’t negated so you make the assumption negated instead and then use the DN rule at the end. It’s also super useful when proving theorems where you have a limited plan of action. An example:
Theorems are formulas that can be proven true without premises so the proofs for theorems have the additional challenge of not being able to build off of premises. Therefore, the above two methods are essential to be able to do proofs of theorems. Here’s an example:
- All assumptions must be discharged(closed).
- Lines between different assumptions must not cross.
- Once discharged, steps within the subproof cannot be used anymore.
On to the next page for a few practice problems!
Hello! I’m aware that I may be a little late since midterms are already over but I have “refound” a channel that I had previously discovered that was a big help to me when it comes to understanding what calculus is, how certain theories and formulas we’re taught in class came to be and what practical problems those formulas and theories were made to solve. This is what your calculus teacher didn’t teach you.
The channel, called 3Blue1Brown, is a channel in the same vein as Numberphile where it tackles mathematical questions but in a very visually stimulating format. For anyone taking calculus (and especially those who’re taking AP Calculus AB/BC), this will help fill in some of the holes in your knowledge on why some things are the way they are.
Different mathematical theories and formulas weren’t invented in a vacuum; each theory and formula was made to solve a problem. So, when learning about math, you shouldn’t learn it in a vacuum either.
Here’s the first episode of a series on the channel, The Essence of Calculus:
Traffic jams are an everyday occurrence especially if you live somewhere decently populated. Sometimes, even though traffic is constantly moving, the speed slows to a crawl out of nowhere. This is especially true when you hit a stoplight and even though it’s a local road during non-rush hour, somehow, there’s a backup of cars for several blocks.
So why does this happen?
As it happens, humans aren’t very patient. Whenever we see space in front of us, we tend to hit the gas to close that distance as fast as possible. However, that also means that we have to stop when that distance is closed. With each start and stop, time is needed to accelerate and decelerate the car. If each car takes just two seconds to accelerate and decelerate, then with just ten cars, there would be a twenty-second delay between the light turning green and the last car starting to move. Now imagine this happening all the way down a highway.
So how do you fix this? First, share this so that more people are aware and not make this mistake and second, try to maintain a constant slow speed whenever you find yourself in a jam. Even if there is not a traffic jam, try not to accelerate too much at one time. Remember, whenever you start and stop abruptly, this effect is multiplied in the cars behind you.
Of course, traffic jams can also be caused by car accidents or road repairs and things like that but they can still be made worse by this behavior. So instead of skipping the line and trying to fit back in right before the point of merging, just take your place in the line and maintain as constant a speed as possible and trying to minimise the number of complete stops you make. If enough people do this, our traffic flow all around the world will become so much smoother.
Oh and the triangle islands at intersections really do help. Instead of having to make a 90-degree turn, cars can merge directly into traffic on a perpendicular road. First, it’s safer and second, it frees up a lane and third, it also doesn’t slow down incoming traffic as much. Smart planning by the city council is just as important as the people using the roads themselves.
That’s all for this time. I’ll talk to you later.
I found a video on YouTube (by ASAPScience) that explains this along with several more details:
One is a magic number. It is probably the first number we learn and in learning it, it opens up a whole new world. However, there is more to one than just the first number we use to count. It is the beginning of everything and it denotes individualism and uniqueness. One is one step from nothingness (zero) and one step from plurality (2). But that’s all nice and ordinary. There’s much more that one can be used for.
Let’s start with an example that a lot of people are familiar with. With those who have done algebra before (which, I assume, is most of you), the power of one is readily apparent. By multiplying/adding/subtracting/dividing one side of an equation and doing the same to the other side with the same value, you can manipulate numbers and sets of numbers into whichever shape and form you want.
This works because you’re changing the values on either side in equal measure, thus maintaining equality on both sides of the equation. This means that, in essence, any equation is just 1=1 (which, when you think about it, is quite obvious).
One also has the unique property of being able to be raised to any power and still remain itself. The same applies to when you take its root (no complex numbers here). One is also the quotient of any number divided by itself. There are many more of these properties that you’ve no doubt learned in your algebra classes.
The main topic today, however, is more interesting. We’re talking about wormholes. Like black holes, wormholes have never been observed directly but there has been evidence of their influence and the math also seems to back their existence (many other things seem to say that wormholes can’t exist). Basically, a wormhole is when spacetime is “folded” over to create a shortcut over long distances. The passage that forms is called the Einstein-Rosen Bridge.
But even if wormholes could exist, it would be highly unstable and too radioactive to interact with. So, let’s just say that, for the sake of argument, we found a way around the radioactive part of the problem and managed to create a wormhole that is big enough for us to get through (which in of itself is a big problem since whatever wormholes do exist are super tiny and they close right after they open). The reason that these guys can’t stay open for longer periods of time (besides the fact that they shouldn’t exist because they violate a whole bunch of rules about the universe) is because of the implosive pressure exerted on it. Now, I know I’m being super vague, but all this stuff is pretty complicated so I’ll just link below to a couple of websites that contain more substantial information.
To counteract this pressure, you need negative mass to maintain a wormhole for any meaningful length of time. This is not to be confused with negative matter. Negative mass is different from negative matter since negative matter still has positive mass.
So, where can we get our hands on something that has negative mass? Well, the answer lies in the field of quantum mechanics (Get it? Field? anyway…). Along with all the other crazy stuff happening in the microuniverse, there exists “exotic” particles. Spontaneously, all over the universe, random particles pop into existence and then disappear. When these particles pop into existence, another particle that is the exact same except with negative properties also form. Then, these two particles pop back together and it disappears. This is where all of this tie back into the concept of one I was talking about earlier. Since the negative particle and the positive particle have exactly opposite properties, it’s like adding -1 and +1 and getting 0. Because of this, these particles don’t technically violate the Law of Conservation of Matter (or energy, depending on what particles you’re using).
However, even if you’re able to obtain enough negative mass (which by itself is already extremely difficult, to say the least), you’d need somewhere to put all the positive mass that you need to conjure up to go along with it in order to maintain the Law of Conservation of Matter. For a wormhole that is the size of a door that stretches the distance of a football field, a Moon’s amount of mass needs to be created and balanced out with its negative counterpart. If you’ve violently jumped into a kiddie pool, then you know what happens when you suddenly displace so much mass/ matter and replace it with something else– only its consequences are far far more far-reaching. But say that we somehow have a way to go around that problem as well. What then?
Now that you’ve created a wormhole, then what? No one knows. If you jump in, you might get ejected into a different time, a different place or a different universe altogether. Or, more likely, you’ll burn up with the radioactivity generated by the mashing of atoms into a space much smaller than what it should be. And that’s not even talking about black holes or white holes. Either way, although we’re able to theorise about such things, we don’t know 100% whether wormholes or black holes actually exist. But perhaps someday, we might have a Tardis of our own and be able to travel through time and space. Until then, I guess we’ll just have to have more physicists digging into the secrets of the universe.
That’s all for this time. I’ll talk to you on Wednesday.
If you guys have any theories or ideas you want to put forth, leave’em in the comments and we might make a post about it. If you have any additional information that you want to share about this topic, tell us! We’d love to hear what y’all know.
Some reference websites