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Elliot:

Can you create a logical expression for that?

(~x∧y)∨(x∧~y)

I thought about it and got a working expression on my second try. I got or(and(not(x),y),and(x,not(y)). Here’s the truth table:

Thinking about the operators as english statements helped.

I’m kind of confused what you’re asking here. The confusion is about “what universal sets can you make” and “what small set of set of operators have you discovered is universal”. Aren’t those the same thing?

To answer what I think you’re asking:

I also went ahead and made “and”, “not”, and “or” out of nor. Assuming that the set {and, or, not} is universal you can make that universal set out of the smaller universal set {nand, nor}.

If x is a million, which side is bigger?

The right side. It means the question becomes: ~which is bigger, 5 million or 10 million?

If x is 0, which side is bigger?

The left. It means the question is just: ~whats bigger 10 or -5?

I find those easy extremes easy.

One thing I am seeing now looking at that equation is that the difference between 5x and 10x is 15, and that x should be a factor of that difference (and of 5). I didn’t really see it like that before. Maybe that’s a bit calculation-y. idk, it could be helpful technique when I estimate other things.

Roughly, how large should x be for the sides to be equal? You can give a range using powers of 10, e.g. between 100 and 1,000. Can you figure out a good range estimate intuitively

withoutany significant calculation?

Glancing at it I think I would say it’s definitely between 1-100. For more precision I think my realisation above would be helpful. Like seeing that x was a factor of 15 means it’s between 1-15. That seems to not be significant calculation to me.

There are also concrete steps that can be used to generate an estimate, which if practiced can create more intuition (and for harder problems, relying more on steps and less on intuition is common).

That sounds cool. I’d like to learn about that. Estimating seems like something that is done a lot in daily life.

If you did the calculation and got the answer 200, would you think that’s a plausible answer or intuitively be suspicious?

I would definitely find that suspicious. I knew intuitively it had to be between 400-500.

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LMD:

So something I’m a bit confused about is what the difference is between making an intuitive estimation, and doing the problem in my head. I find myself looking at an equation like 5x + 10 = 10x - 5 and beginning to solve it in my head with not much difficulty. But it seems like I should be doing something other than that?

This one isn’t hard to solve so you might find it easier to practice on a more complex equation where using explicit exploration steps would be more usual. But you can also examine this one without solving or simplifying it as a way of practicing.

If x is a million, which side is bigger?

If x is 0, which side is bigger?

Roughly, how large should x be for the sides to be equal? You can give a range using powers of 10, e.g. between 100 and 1,000. Can you figure out a good range estimate intuitively *without* any significant calculation?

There are also concrete steps that can be used to generate an estimate, which if practiced can create more intuition (and for harder problems, relying more on steps and less on intuition is common).

Does this make some sense and seem potentially useful?

If you did the calculation and got the answer 200, would you think that’s a plausible answer or intuitively be suspicious?

]]>- make up my own problems
- find linear equations worksheets and estimate the solutions
- research learning estimation in math and associated materials (I did this and didn’t find anything good in the short time I spent looking)

So something I’m a bit confused about is what the difference is between making an intuitive estimation, and doing the problem in my head. I find myself looking at an equation like 5x + 10 = 10x - 5 and beginning to solve it in my head with not much difficulty. But it seems like I should be doing something other than that?

As for graphing something, that’s not something I know how to do yet.

Here is some work I did:

- The value of 3x+2 is 9 plus twice the value of x+5. What is x?

- The number 1341 can be written as the sum of three consecutive positive integers. What is the largest of these integers?

- Ten more than five times x equals five less than ten times x. What is the value of x?

- In 36 years Adam’s age will be 2.5 times his current age. How many years old is he now?

- A tank is to be filled with water. When the tank is one-sixth full, 130 gallons of water are added, making the tank three-fifths full. How many gallons does the tank contain when it is completely full?

Eternity:

So xor just says only one of the statements can be true.

Can you create a logical expression for that?

Ah, so it was just guessing rather than working out a solution.

Yes. So what small set of operators have you discovered is universal?

Also what universal sets can you make from: {and, or, not}?

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Elliot:

Can you describe xor in English?

Uhh. The way I think about it when I see it is to read it as “either-or”, meaning that either, but not both, of the the statements can be true. So xor just says only one of the statements can be true.

Yes. If both are true it’s true, if both are false the statement is still true. Both inputs need to be equal for it to be true. Makes sense.

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Elliot:

I’m not sure if you’re claiming you could do that, or you think you did it, or you had a different goal, or what.

The way I went about answering the question was assuming the below was true:

I assumed that I could come up a smaller set of universal set of logic operators smaller than {and, or, not} since it was asked of me. From there I just assumed that since you can convert between “nor” and “and”/“nand” and “or” you could probably get “nor” and “nand” to do some functions similar to “and” and “or”.

Do these work? (these were made just now after the fact, so I didn’t initially know I could do these):

EDIT:

Hmm. I don’t think I answered your question that well. My goal was just answering the question of finding a smaller universal operator. I guess I was technically claiming I could do that. Yes.

]]>Prediction examples: “linear increasing; solution near 100” or “exponential decreasing; solution near 0”.

Another way to make a prediction is to draw a rough graph and mark the solution. Sometimes this is quite hard but others times it’s not very hard; it depends on the problem.

With experience and looking for them, you can recognize patterns/features in the problem/equation/expression that (usually) correspond to various predictions or graphs.

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Eternity:

xor = not(or(and(x,y), and(~x,~y)). Ok, this one was a failure when it came to thinking about what the operators mean. I got an answer by noticing that if I just did a not on my answer for equality I got the table for xor.

Can you describe xor in English?

If you look at the truth table, can you see how the name “equality” makes more conceptual sense in general?

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Eternity:

Starting with the easier version. Assuming that {and, or, not} is universal you can just use nand or nor. You can convert between “and” and “or”. From what you had me do earlier with De Morgan’s laws: or(x,y) = not(and(not(x),not(y)), and(x,y) = not(or(not(x),not(y)). I think you can negate both sides, which gives you: not(or(x,y)) = and(not(x),not(y)) or nor(x,y) = and(not(x),not(y)). Also not(and(x,y)) = or(not(x),not(y)) or nand(x,y) = or(not(x),(y)).

I’m confused. Can you do:

and(x,y) = ?

not(x) = ?

using only nand on the right?

I’m not sure if you’re claiming you could do that, or you think you did it, or you had a different goal, or what.

]]>]]>I find this is also true in my experience writing and playing music with people.

I’ve basically had no luck creating the basic musical ideas of a piece in a group setting. Very little to no progress was made in the projects I was involved in as a writer in this way. It always ended up being more efficient to just work on new ideas alone, and then bring them to a group to develop, flesh out, whatever. Sometimes even that was hard and basically everything would be done alone, and then brought to the group to consider/learn. Efficient is not even the right word, there was basically no chance of having something to work on if something wasn’t developed to a certain point alone. This situation had made me think over the years that I just wasn’t good at collaborating with people in general.

So it’s interesting to hear that others are finding this kind of collaboration unproductive too.

Some of the problems I encountered in a music context were:

- Multiple instruments in a room trying to make different things will create a lot of noise and distractions, so you kind of need to be all working on the same thing.
- Following a lead on an idea can take time, lead nowhere, and sometimes/often needs the participation of others. This can make you risk averse and passive to avoid wasting others time. This can block creativity. What has ended up happening is that some people become more passive than others and then the others feel more pressure.
- You feel a need to be working on something because that’s what you’re all there to do but there are no promising ideas. This makes it awkward and feel urgent that there be something to work on. This is bad for creativity. This also makes you just want to call it a day. it sucks.
So I ended up doing the vast majority of my work on music projects on my own, and it improved lots of things. People understood my ideas better because they were more developed, they could better judge whether it was worth it to them to spend more of their time on it and learn it, we had clearer things to work on, and more things to work on. I had less of the experience of getting stuck and calling it a day.

It also makes me realise that I haven’t really tried to make music with others in an asynchronous, remote, online-group-type way.

more exploring:

This one I couldn’t remember the original answer so was legitimately estimating. It took 7 steps before it was right:

Some before I had solved the problem:

- What is the number that yields the same value when it is multiplied by three and then increased by five as when it is multiplied by five then decreased by three?

- Joe and Mary each pay the same price for a T-shirt. Before they buy it, Mary has \$ 2 more than Joe. To buy the T-shirt, Joe spends \$1 less than \frac{2}{5} of his money, and Mary spends \frac{1}{3} of her money. What was the total amount of money Joe and Mary originally had altogether?

So from nand or nor you can get the three universal operators {and, or, not}. I think I answered the below:

Mmm. If those set of logic operators can be made to do everything more specific logic operators can do?

EDIT:

~12 minutes.

Can you come up with a universal set of logic operators that is smaller than {and, or, not}? How can you determine if a set of logic operators is universal (both in general or the easier version: using as a premise that {and, or, not} is universal)?

15-60min

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Elliot:

Assignment: Using {and, or, not}, construct xor, nor, implies and equality.

I spent roughly an hour. Half-way through I realized I wasn’t timing. Thinking of the phrases in terms of English sentences helped. I did that in conjunction with creating a bunch of tables.

xor = not(or(and(x,y), and(~x,~y)). Ok, this one was a failure when it came to thinking about what the operators mean. I got an answer by noticing that if I just did a not on my answer for equality I got the table for xor.

Now that I think about it. I guess that makes sense? I got my answer for equality (I did have to look up what that meant and learned that it was the same as the bi-conditional I learned so far) by thinking about how the table is saying both have to be true or both to be false for the statement to be true. The negation of that would be only one can be true and false.

nor = not(or(x,y))

implies = or(~x,y). I didn’t have a chance to think about this one. I just noticed they matched.

*

equality = or(and(x,y), and(~x,~y)

Says basically working in in-person groups sucks but working in async, remote, online groups is good.

]]>1960s protests convinced politicians to make college expensive. America is still paying the price

LMD:

My intuition would say that the ‘result’ (double the number) was over 20, but not necessarily that the number (n) was more or less than 20.

OK so intuition for numbers can be improved with practice. Here is a way to practice it for this problem and many others. This is often best to do before you start solving the problem, but can be done later too.

Take the problem and try a low, medium and high number. (For multi-variable problems, you can do low,low, low,medium, etc. and get 9 pairs, or can do 4 pairs with only low and high, or can keep one unchanging while changing the other variable. There are more options.)

So in this case, for medium I’d pick 20. For low, I’d pick 2, but 10 would be fine as well. The reason for 2 is that negative numbers, 0 and 1 can all be special cases, so I just picked the lowest integer that avoids all that. For high, 30, 40 or 100 would all be reasonable choices (representing: add 10, double the 20, or use a number that’s significantly higher). It can be good to have the numbers evenly spaced (e.g. use 10/20/30) but isn’t needed. You can try additional numbers if you think they’ll provide any useful information.

Trying out some numbers in problems can help you get a feel for what the answer may be like (high, low, medium, and other things like negative or positive, or between two numbers).

You can find patterns like as you increase the inputs, the outputs increase. Combine that with knowing that an input of 20 gives a result that’s too low, and you know the solution must be more than 20.

You can quickly narrow the solution down, like if you also try 100 and the output is too big, then the solution is between 20 and 100. Then you can try the midpoint of the range, 60, and narrow it down to a range of 40. For problems with integer solutions, you can actually get the exact answer just by checking the midpoint of the range repeatedly without it taking a ton of steps. Like narrowing down from 80 possible solutions to 1 is doable within 7 steps.

The main point here is to explore the problem by finding out basic information like whether higher inputs cause lower or higher outputs, and whether the change is linear or exponential, and narrowing things down like the answer needs to be big, small or negative. If you do this, it may help you solve the problems, and it’ll definitely help you intuitively understand what’s going on more and give you an extra way to check for errors (know if your answer is reasonable or not).

So try this on the coins problem and some others.

]]>- Joe multiplies a number by 4, adds 1, and then divides by 3, getting a result of 7. Sue divides the same original number by 3, adds 1, and multiplies by 4. What result does she get? Express your answer as a common fraction.

- The \$4.55 in Carol’s piggy bank consists of quarters and nickels. There are seven more nickels than quarters. How many nickels does Carol have in her bank?

- Six friends went to a restaurant and agreed to share the bill equally. However, two people forgot their wallets so the other four friends’ portions of the bill went up by \$7 each. How many dollars was the total bill?

- The bottom cup in a 73 cm tall stack of nested identical cups is 10 cm tall. If each additional cup adds 1.5 cm to the height of the stack, how many total cups are in the stack?

- One caterer charges a basic fee of \$100 plus \$15 per person. A second caterer charges a basic fee of \$200 plus \$12 per person. What is the least number of people for which the second caterer is cheaper?

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]]>Are people who smoke pot the same as people who steal? In other words: are people who do bad things to themselves the same as people who violate rights? A lot of people would call both group of people immoral. Sure, I guess. I don’t know though. I’ve thought of this issue a few times and its weird every time I think about it. Like sure people who spend their days drinking, smoking pot, eating junk food, and all that are not really living that much of a life. I know some people would brand such behaviors as immoral and bad. I more-or-less feel that way too about those things (though I’m by no means perfect in living my life), but I think sometimes the lines get blurred. Sure I think smoking pot is bad, but is it the same bad as someone who steals? Both, by many people, kinds of people would get categorized as immoral but only one I feel deserves to be treated as “bad” for their actions. Yeah spending all day smoking pot is lame and pretty bad but I don’t think a pothead should be treated the same as someone who’s a thief. It’s a weird blurred line.

~Huh. Now that I think about it: this may have some kind of relation to the multi-factor decision making article from Elliot. Specifically the section: “Converting Dimensions to Goodness”.

Mmm. Yeah. Kinda? Idk. Too tired right now, but looking through that section. Goodness is kinda vague and people just arbitrarily convert things to how good they are. Different things have different ways of being good in different contexts. Bad things are similar.

- Jeff has an equal number of nickels, dimes and quarters, worth a total of $1.20. Anne has one more of each type of coin than Jeff has. How many coins does Anne have?

LMD:

My intuition would say that the ‘result’ (double the number) was over 20, but not necessarily that the number (n) was more or less than 20.

Sorry as in, the result of adding twenty to a third of a number is more than twenty.

]]>
Elliot:

Why didn’t you check your solution for that one?

I’m not sure.

One thing was that I was focusing more on sections 1 and 2. I was also doing these in a short window of time before I had to leave for work, and I wanted to get through the 5 I had selected. Problem 4 seemed to me to be very short and not offer much to practise re sections 1 and 2, so I might’ve been keen to just move on to the next problem (which I didn’t end up getting through in time). That could be why I skipped checking it.

But, I probably also just intuitively thought that was the right answer and so not thought I should check it. But using intuitions about when to check your work that rely on intuitions about whether your answer is right seems problematic though now I say it.

My intuition would say that the ‘result’ (double the number) was over 20, but not necessarily that the number (n) was more or less than 20.

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Eternity:

I spend ~30 minutes doing this before I realized. Is it fine the way I’m approaching this so far? I don’t necessarily mind trying out various arrangements until one gives me the table I need, since this is all new to me. I don’t think that’s what I’m supposed to be doing though, but I could be wrong. Figured I’d ask.

It depends. Sometimes when people work on stuff, even in less productive or more brute-force-style ways, they start noticing some patterns and developing intuition after a while, just not right away. But sometimes they don’t. How well it works depends on their attitude, background knowledge, how long they spend on it, and more.

If you don’t mind doing more then it’s probably OK. But here’s another way to approach it if you prefer:

You can translate logic expressions into English sentences describing them (e.g. “or” = “at least one is true”). And you can translate from English to other English sentences that are different but still logically the same. And you can translate from English to logical expressions. E.g. “at least one is true” = “not both are false” = “!(!x & !y)”. Using English can help you think conceptually about stuff instead of brute forcing. But you’ve tended to do better at non-word math problems than word math problems so it might not work well for you.

There’s also a generic way to construct an expression to achieve any truth table you want, using only {not, and, or}. I wouldn’t expect you to figure it out right now but it’s not super complicated so it’s not out of the question.

Overall, one of the main goals here is to build up more familiarity and intuition with logical operations (and also at some point to connect them more to English sentences so they can be used in philosophy activities like text analysis, debate, etc.)

]]>The only one I know here is nor = not or.

]]>
Elliot:

you got it. i didn’t say you’d use one of each. try it with nand and nor too.

Those two work as negation. Which makes sense since not and = nand and not or = nor. So when its just two inputs “and” and “or” work as the identity and nand and nor work as the negation.

Ok. So I got an answer today. I forgot that I could just negate the variable(?) itself. So I wanted to give it a quick try before dropping it. I spent like ~10 minutes. I saw that and(not(x),not(y)) gave me

and so I just did not on that whole thing and got:

.

Here’s the full table:

After that I assumed that the same process would work with going from and to or. So here’s that table:

EDIT:

Tbh. I don’t know if I really understood anything from doing this. Since it was just coming up with a bunch of tables until one worked.

Eternity:

Hmmm. I guess there’s some background information I’m not understanding here. I thought all this made sense to me so far.

I don’t think most people ever reinvent the laws. I thought it was worth a try. Let’s try this next (there’s a chance it’s too hard but you did the same thing with nand so worth a try):

It turns out that {and, or, not} is a universal set of logic operators (this is closely related to stuff you may have heard about computational universality). You can construct all the other operators from them.

Assignment: Using {and, or, not}, construct xor, nor, implies and equality.

Max 60min.

PS This is standard logic. There are other types of logic (that you don’t need to know about) including reversible logic, where, given the outputs and the arrangement of operators used, you can figure out what the inputs were. Reversible logic is relevant to quantum computing and to computing with low energy usage.

]]>
Eternity:

Hmm. From the resources I’ve looked at so far I learned about four unary operations. Always true, always false, identity, and negation. I assume one is always true, zero is always false, not is negation, and identity is identity.

yes

you got it. i didn’t say you’d use one of each. try it with nand and nor too.

]]>Hmmm. I guess there’s some background information I’m not understanding here. I thought all this made sense to me so far.

Oh yeah. Haven’t looked up anything during this.

]]>
Elliot:

Oh and if you understand then label which are which. For {or, and, xor, implies} label each one with one of {one, zero, not, identity} for how it behaves when given 2 of the same input.

Hmm. From the resources I’ve looked at so far I learned about four unary operations. Always true, always false, identity, and negation. I assume one is always true, zero is always false, not is negation, and identity is identity.

Looking at the truth tables:

One is always true. One is implies.

Zero is always false. Zero is xor.

Identity is the same as the input. Hmmm. “or” and “and” seem to satisfy this.

Negation is the opposite of the input. Hmmm. Nothing seems to satisfy this.

Did I mess up somewhere?

or (x,x) if x is true. Then its true. If x is false. Its false.

and (x,x). If x is true. Then its true. If x is false. Its false.

Uhh. Hmm. Lets check the other ones:

xor(x,x). If x is true, then both are true. Not either or. False. If x is false, then both are false.

implies (x,x)

If x is true its true. If x is false its true.

Hmmm. What am I missing?

]]>
LMD:

I got the problems from here in Eternity’s tutoring thread.

All my problems are from Alcumus. You seem to have used it before:

Where it says focus, you can click on change focus, and choose a topic area to focus on.

If you get enough right it will move you away to another topic but you can always just reset it back to word problems.

EDIT:

They also have difficulty selection in the settings:

Elliot:

Small thing: you used the number 1.5 instead of train_b_late_start_hours a few times. The basic downside of that is the number isn’t self-explanatory.

Gotcha.

Here are some simple ones that I did yesterday with times. I got the problems from here in Eternity’s tutoring thread. I haven’t checked the answers yet, and there’s one more I need to do when I’ve got time. I haven’t done another rate one yet but I’ll find some more asap.

- A girl is half as old as her sister and is also two years younger than her brother. If the sum of the three children’s ages is 34 years, what is the product of their ages?

- In a certain school, 2/3 of the faculty are women, 1/4 of the men on the faculty are married and the other 9 men are bachelors. How many faculty members are there?

- John divided his souvenir hat pins into two piles. The two piles had an equal number of pins. He gave his brother one-half of one-third of one pile. John had 66 pins left. How many pins did John originally have?

- If 20 is added to one-third of a number, the result is the double of the number. What is the number?

A refutation is a type of IGC triple, yeah. The goal is to criticize a different IGC. Treating criticism explicitly as IGC triples can sometimes be unnecessary/skipped. Having a few core concepts that are used broadly, instead of a bunch of different concepts, is relevant for AGI design (fewer data structures; more elegant code with fewer special cases; also IMO if you can’t figure out how to evolve ideas that function as both solutions *and* as criticism of other ideas, using the same data type, then you probably aren’t getting anywhere on AGI. The same idea data type should also function as goals or contextual information. Or put another way, I think you should just have one idea replicator not several different replicators for different purposes).

CF denies that you can do criticism (in the CR error correction sense) independent of goal(s) and context. Errors cannot be evaluated without some sort of contextual information about what success and failure are.

Synonyms for goal, like “objective” or Popper’s “problem”, are fine instead. Considering the goal to be part of the context, or vice versa, can work too. It’s also possible to mix the goal and/or context into the idea too. Like you could say proper ideas are IGCs. Not wrong, but not my general recommendation. Depending on the context, the “idea” part of IGC can be better called something more specific, like plan or solution; using the very broad word “idea” to mean something a bit closer to “solution” has some downsides but overall I find it works pretty well. This stuff connects most with CR but also relates to Objectivism’s claim about knowledge being contextual and to Theory of Constraint’s material on goals.

Yeah I agree. I see how that could be confusing. CF basically says it’s valid to criticize things for *not being good enough* – you can (and often should) demand more from ideas. This comes up a lot in debates/discussions where people concede too early/easily without being demanding enough to get full information about why the new ideas, that they’re (partially) losing the debate to, are good enough to address all their relevant goals/problems. Lots of times people try to change their mind because of some arguments showing a new idea is *better* (by degree), instead of them actually insisting on understanding how to use it to solve all relevant problems before adopting it. (If their own idea doesn’t meet that standard due to criticism they were just told, they should take the position that they don’t yet know a full solution, can use parts of both their old and new ideas in the mean time, and can do more research.)

In this case, not going to the store isn’t adequate as a positive solution to getting food. That does *not* mean all options that qualify as “not going to the store” are wrong (which would lead to the logical problem); it just means that the positive knowledge/idea “don’t go to the store” is inadequate to solve the problem. It’s not categorically wrong but it doesn’t offer enough help to solve the problem.

I think I may have been interpreting some of what CF says in a sense that is more philosophically fundamental than how you actually meant it. You use very simple language in your articles, and I have been reading things between the lines that maybe aren’t there.

Was I at least right that CF treats a refutation (a reason why an idea fails at a goal) as being an IGC triple itself? Or is a refutation something else? Or does CF not take a stand about that?

I encountered something I found confusing about CF. I think I figured it out, but I’ll write it out anyway. It’s related to the above question, because it pertains to CF’s nonstandard view of refutations.

Let

I_1 := “going to the jewelry store on 5th ave.”

I_2 := “*not* going to the jewelry store on 5th ave.”

G := “getting food.”

C := some typical context. A normal adult human in NYC who is hungry.

Clearly (I_1, G, C) is refuted, by the criticism that jewelry stores don’t sell food.

But *also*, (I_2, G, C) is refuted, because the idea of not doing some specific action isn’t sufficient to meet the positive goal of getting food. To satisfy G, you’d have to add something to I_2, like going to the grocery store on 7th ave instead.

Someone could summarize this situation (and CF often shortens things like this) by saying: *going* is refuted, and also *not going* is refuted. And if you’re not careful, that summary *looks like* it violates the law of excluded middle; it could be misconstrued as meaning that a *proposition* (“It is the case that I should go to the jewelry store on 5th ave”) and its denial (“It is *not* the case that I should go to the jewelry store on 5th ave”) are both refuted.

Note: this is a system of linear equations where you’re using around 7 equations. But it’s not actually much harder to calculate than a system of 2 linear equations. That’s related to the equations mostly being simple and most of them having a low number of variables.

Switching the train direction doesn’t take many changes when the problem is set up well.

da0 + (sa * t) = db0 + sb(t - 1.5)

IMO this goes in section 3, not 2. I consider it a calculation based on equations from section 2.

Anyway, that’s good. Try some more word problems and hopefully this will be helpful for future problems.

]]>I really struggled with this and I don’t know why yet. I think I was second guessing myself a lot. Maybe I wasn’t focusing enough on my understanding of the problem and my ideas on how I could solve it, and was focusing on trying to interpret or decipher the clues given to me. (That’s why I asked that clarifying question before) And also I think I was assuming a lot about what kind of result I should get. Like I thought I should have two linear equations with two unknowns each.

I also didn’t find writing out the long variable names in the equations very helpful. I found it made some things harder. Also hard to like, see the equation when the operators were so far apart. Something I did find helpful (which is similar) was writing out in words what each part of an equation meant for example:

sb(t-1.5) = distance travelled by b in 1.5 hours less than t

sa = 55

sb = 130

train_b_late_start_hours = 1.5

t > 1.5?

da0 = 360

db0 = 0

dat = da0 + (sa * t)

dbt = db0 + sb(t - 1.5)

dat = dbt

da0 + (sa * t) = db0 + sb(t - 1.5)

360 + 55t = 0 + 130t - 195

555 = 75t

555/75 = t

t = 7.4

sa * t = distance of train a from station A when the trains meet

55 * 7.4 = 407km

OK

sa = 55

sb = 130

train_b_late_start_hours = 1.5

t > 1.5?

da0 = 360

db0 = 0

dat = da0 - (sa * t) - *changed*

dbt = db0 + sb(t - 1.5)

dat = dbt

da0 - (sa * t) = db0 + sb(t - 1.5) - *changed*

360 - 55t = 0 + 130t - 195

555 = 185t

555/185 = t

t = 3 hours until trains meet

sa * t = 165km from station A

dat = da0 *minus* (sa * t) - because train a is travelling in the opposite direction, so the distance it travels in time t is negative.

Elliot:

They don’t have any more information available than when receiving 1 input, so they actually have to give the same outputs as one of the functions that takes only 1 input. There are only 4 binary functions of 1 input: one, zero, not, identity which correspond to the 4 possible truth tables for a function with 1 input (10, 11, 00, 01 are the only 4 options). (The one and zero functions don’t actually need any input because the output is the same regardless of the input.)

Oh and if you understand then label which are which. For {or, and, xor, implies} label each one with one of {one, zero, not, identity} for how it behaves when given 2 of the same input.

]]>They don’t have any more information available than when receiving 1 input, so they actually have to give the same outputs as one of the functions that takes only 1 input. There are only 4 binary functions of 1 input: one, zero, not, identity which correspond to the 4 possible truth tables for a function with 1 input (10, 11, 00, 01 are the only 4 options). (The one and zero functions don’t actually need any input because the output is the same regardless of the input.)

Make sense?

If so:

Next you can try to reinvent De Morgan’s two laws. They’re for converting between “and” and “or”.

Law 1:

or(x,y) = ?

Answer only using: and, not

Law 2

and(x,y) = ?

Answer only using: or, not

Max 60min.

]]>EDIT:

Like this?

In general I’d say I do have a really bad problem of trying to figure out what the teacher wants. Its partially why I really hate school and especially any kind of writing/free-response assignments in school. They are so open-ended that I get extremely stressed trying to figure out what the teacher is looking for. I think I’ve dropped more classes due to the broadness of free-response assignments than anything else (why I really hate english class and essays). I think this is also, partially, why the only classes I’ve done consistently well in school is math.

After re-reading it:

Since verbal behavior (spoken or written) is what gets the gold star, students begin to think that verbal behavior has a truth-value.

Hmm. I’m sure there’s plenty of stuff I’ve learned in school that are just noises, or pictures in the case of writing, that are meant to satisfy teachers. However, I don’t think that has been my particular issue when going through school. I’ve never fully felt comfortable in saying words to satisfy a teacher. I’ve always tried to figure out what the stuff I’m saying means. I always felt like if I’m just repeating without understanding I’d be lying. “Surely the teacher will know that I don’t understand what I’m talking about.” This mentality has lead to a huge deal of stress because I typically end up trying to figure out too much, overreach, burn out, and quit.

I guess partially I have that mindset? I don’t try to necessarily find the vibrations to satisfy my teachers. In general, my experience through school has been that I’ve typically been the one to best understand a subject and connect it with my experiences compared to a lot of peers (and I think some teachers too). How often I put the effort in and enjoyed the material to do such a thing is different.

To reply to what you’ve said: yeah, I do think too much of what my teacher wants.

]]>Make a truth table with the following columns (I’ll use s-expression notation this time to emphasize that content and notation are independent):

x

(identity x)

(not x)

(or x x)

(and x x)

(xor x x)

(implies x x)

Elliot:

Make and compare truth tables for or(x,x) and or(x,y). Do them as two separate tables, not in the same table.

For or(x,x) with identical inputs the output is the same as the input. Putting it into a statement: it will rain or it will rain. Its the same as only dealing with just it will rain and whether thats true or not.

For or(x,y) when x and y are both true or both false they have the same output as or (x,x). That makes sense. Since when both are true and when both are false they have the same input.

So or (x,y) is the same as or(x,x), or or(y,y), when the x and y represent the same thing.

]]>
Eternity:

I did think while doing it that whatever I come up with would be equivalent to not (and (x,y)). I just didn’t think that would be a satisfactory answer because it said you can use or and because the long time frame made it seem like this would take a bit time to think of and do.

I didn’t say you had to use or. You may have some of this mindset where you’re thinking too much about what I want instead of the topic itself: https://www.lesswrong.com/posts/NMoLJuDJEms7Ku9XS/guessing-the-teacher-s-password

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