Math with the passion of a THOUSAND SUNS

[rong]

so, you're saying, in a situation where you wanted to distinguish the set {can} from the set {can, can, can} you could (or would) describe the {can, can, can} set as {{can,1},{can,2},{can,3}}?

[Golden Applesauce]

so, you're saying, in a situation where you wanted to distinguish the set collection {can} from the set collection {can, can, can} you could (or would) describe the {can, can, can} set collection as {{can,1},{can,2},{can,3}}?

Yes. I would write it as [can, can, can], though. The [ ] brackets haven't gained wide acceptance in math literature that I know of, but Python, Ruby, and JavaScript all use them as notation for quickly and easily writing lists. I would never drop down to the level of the set theoretic definition unless I was actually writing a proof about a basic propety of lists. Lists are well understood enough as their own kind of object that you don't need to go into that level of detail. Even if you do, it's more common to treat the list as a function. Something like:

f (n), for n in { 1, 2, 3 }
= can, for n=1,
= can, for n=2,
= can, for n=3

Except I would start numbering from zero, of course.

A mathematician would never actually write { can, can, can } except to make a point. As a set, it is 100% literally indentically the same entity as the set { can }. To even talk about the two notations referring to different objects you have to use a word other than 'set', which has a very specific mathematical definition.

[rong]

oh, right - i see my goof - you can't distinguish the set {can} from the set {can,can,can} because they are the same - that would be like trying to distinguish 1 from 1, right?

Unions and Intersections of sets are something that seems kinda familiar - I can't seem to remember if there are any other set theoretic operations, but now I'm thinking that:
{can,1}U{can,2}U{can,3}={{can,1},{can,2},{can,3}}  (where "U" represents "Union")

but that has me thinking that {{can,1},{can,2},{can,3}}={can,1,2,3} - if this is the case, then I don't see a way to make the jump from sets to lists.

whenever I see [ ] brackets, I think vectors or matrices -which are ordered lists.

I apologize if I'm jacking this thread into somewhere you didn''t intend it to go - I realize I could probably research all this on my own - but I'm enjoying it as a conversation and hope you are too.

[Golden Applesauce]

oh, right - i see my goof - you can't distinguish the set {can} from the set {can,can,can} because they are the same - that would be like trying to distinguish 1 from 1, right?

Exactly right - just like 1 and 1.0 are the same number.

Unions and Intersections of sets are something that seems kinda familiar - I can't seem to remember if there are any other set theoretic operations, but now I'm thinking that:
{can,1}U{can,2}U{can,3}={{can,1},{can,2},{can,3}}  (where "U" represents "Union")

but that has me thinking that {{can,1},{can,2},{can,3}}={can,1,2,3} - if this is the case, then I don't see a way to make the jump from sets to lists.

whenever I see [ ] brackets, I think vectors or matrices -which are ordered lists.

I apologize if I'm jacking this thread into somewhere you didn''t intend it to go - I realize I could probably research all this on my own - but I'm enjoying it as a conversation and hope you are too.

(going to try using underlines on nested sets, should make them easier to see visually)

{ { can, 1 }, { can, 2 }, { can, 3 } } is a different set from { can, 1, 2, 3 }. You can tell because the first one has three elements, which are all sets, and the second one has four elements: a can, the number 1, the number 2, and the number 3. The elements are different, therefore the sets are different.

I think you're confusing union with composition. The union of two sets A and B is a new set C with the property that everything in A is in C, everything in B is in C, and nothing else is in C. The union of { can, 1 } U { can, 2 } U { can, 3 } is { can, 1, 2, 3 }. All four of the elements in the result set are in at least one of the three original sets, and every element of an original set is in the result set.

The composition of two sets A and B is a new set with exactly two elements: the original sets A and B. So composition(A, B) = { A, B } = { A } U { B } ("union of the set containing only A and the set containing only B") which is probably not equal to A U B ("union of the sets A and B"). Composing { can, 1 }, { can, 2 }, { can, 3 } together would give you the set { { can, 1 }, { can, 2 }, { can, 3 } }

I'll delay the Balrog a bit and do a full post on set operations. Union/Intersection probably make more sense to learn first, anyway. Making the jump from sets to lists doesn't require much more, so maybe I'll do that next if people don't have too many questions on first-order predicate logic.

[rong]

ok, i guess i arrived at the correct {can,1}U{can,2}U{can,3}={can,1,2,3} by using the incorrect: {can,1}U{can,2}U{can,3}={{can,1},{can,2},{can,3}}

i'm a little fuzzy on this:

composition(A, B) = { A, B } = { A } U { B } ("union of the set containing only A and the set containing only B") which is probably not equal to A U B ("union of the sets A and B"). Composing { can, 1 }, { can, 2 }, { can, 3 } together would give you the set { { can, 1 }, { can, 2 }, { can, 3 } }

i will try an example:
let A = { can, 1 }
let B = { can, 2 }

then
A U B = { can, 1 } U { can, 2} = { can, 1, can, 2 } = { can, 1, 2 }
composition(A,B) = { A, B } = { {can, 1}, {can, 2} }
{ A } U { B } = { A, B } = { {can, 1}, { can, 2 } }

[side note] I'm starting to think that all you have to do is replace any " } U { " with  a " , " (comma)
of course, this kind of thinking reduces any reasoning and thought to simple symbol manipulation - which i think mathematicians despise.  [/side note]

although i think my examples are correct (they duplicate what you said, but I promise i thought them through on my own), when I review, I realize my difficulty lies in the distinction between A and { A } - i.e. it seems to me that A = { A } should be true, but i am realizing that is not the case.

working through the example again:

let A = { can, 1 }
let B = { can, 2 }

then

{ A } = { {can, 1} }
{ B } = { {can, 2} }

A U B = { can, 1 } U { can, 2} = { can, 1, can, 2 } = { can, 1, 2 }

composition(A, B) = { A } U { B } = { {can, 1} } U { {can, 2} }  = { {can, 1} , { can, 2} } = { A, B }

i think i got it now.  it certainly appears that " } U { " = " , " seems to hold.  I'd be interested to see an example where it fails.

i'm not sure if it's the right use of the term, but i think the real check of understanding would be with "nested" sets.

example:
let A = { can, 1, { can, 2 } }
let B = { can, 3, toucan sam }

then:
A U B =  { can, 1, { can, 2 } } U { can, 3, toucan sam } = { can, 1, { can, 2}, can, 3, toucan sam } = { can, 1, 3, toucan sam, { can, 2 } }

composition(A, B) = { A } U { B } = { { can, 1, { can, 2 } } } U { { can, 3, toucan sam } } = {  { can, 1, { can, 2 } } , { can, 3, toucan sam } } = { A, B }

is there a term for levels of "nesting" in sets?  would this be 1st or 2nd order sets or something? (i.e. A is 1st order, B is second order) it seems like there should be something in the definition of an operation that addresses this?  maybe not?

[Golden Applesauce]

I'm starting to think that all you have to do is replace any " } U { " with  a " , " (comma)
of course, this kind of thinking reduces any reasoning and thought to simple symbol manipulation - which i think mathematicians despise.

On the contrary - mathematicians love it when they can reduce thinking to the level of simple symbol manipulation. They just don't like doing lots of symbol manipulation. If they can reduce something to symbol manipulation, they can safely let computers and undergrads to do all of their work for them. Then they've beaten that particular field and move on to something more exciting.

is there a term for levels of "nesting" in sets?  would this be 1st or 2nd order sets or something? (i.e. A is 1st order, B is second order) it seems like there should be something in the definition of an operation that addresses this?  maybe not?

Sort of. I remember doing proofs with the concept, but not what we were proving. I'll check my textbooks.

Most of the time, we don't care whether a set contains other sets or contains things that aren't sets. That's because all sets end up falling into two categories:
1. The empty set, which contains nothing.
2. Sets that contain other sets.
i.e., in pure set theory, everything that exists is a set - even numbers and functions - so it isn't meaningful to talk about whether a set contains sets or not-sets.

[Golden Applesauce]

I'm a bit of a pervert, so lets talk about set operations in terms of

PREDICATE LOGIC


I'm going to assume you're already familiar with ordinary basic logic ("Dogs are blue AND 3 is prime", etc), understand how to tell whether two sets are actually the same set (covered in the previous big post - if it still doesn't quite make sense, ask, I can give more examples), and can use the English word "is" pretty well. If you understand those three things and you leave this post without everything making perfect sense, ask me questions.

The 'predicate' in 'predicate Logic' is the same predicate you learned in elementary school when you had to divide English sentences into their subject and a predicate. The subject is what the sentence is about, and the predicate says something about the subject. Some form of the copula ('is', 'are', 'to be') is usually present.

In the sentence:
Roses are red.
'Roses' is the subject. We are making a statement about roses. 'are red' is the predicate. We are saying that something ('roses') satisfies a condition ('are red'). We are not saying that roses and red are the same concept, that you could paint a fence with a bucket of roses paint or that the color red is a fragrant, thorny flower. This is the 'is of description', not the 'is of identity'. English confuses those some times, but you all already know all about e-prime so lets fast forward through that.

[tangent] Did you know that not all languages are subject-predicate based like English? Japanese, for instance, uses a topic-comment organizational structure instead. The classic sentence illustrating this is Boku wa unagi da. wa is the topic marker, which means that the word before it, boku, is the topic. boku is a mildly humble way for a male to refer to himself. unagi is eel. da is the copula. Google Translate translates this as 'I is eel.' We could clean that up to "I am an eel", which is a predicate ('am an eel') about a subject ('I'). But that's not what boku wa unagi da means at all - the speaker isn't an eel, he's a person ordering food. boku, 'I', is the topic, not the subject, and 'eel' is a comment, not a predicate. What just happened is that the previous six people all ordered sea urchin sushi, so when it's his turn to order he says boku wa to change the topic to himself and then unagi da to comment that he wants to eat eel sushi. A better translation would be 'For me - eel.'
[/tangent]

What makes predicate logic predicate-based is that we can divide statements up into their subject and predicate and treat the predicate as an entity in its own right. We can split out the 'are red' from the rest of the sentence and do stuff with it, and we can substitute things other than 'roses' in for the subject. That's more flexible than than simple propositional logic, which treats statements like '3 is prime' as indivisible units.

notation break: Mathematicians realllly hate the English language, so when they make subject-predicate sentences they do it in their own funny language that they claim is less confusing. In English, we would say:
Roses are red.
to apply the predicate ('are red') to a subject ('roses'). Mathematicians like to reverse the order and write:
is_red(roses)
instead, where is_red is the name they gave to the 'are red' predicate. If you think that looks a lot like the notation for a function, you are ahead of the game: predicates can be thought of as functions that only map to True or False.

In ordinary propositional logic, you can only view propositions at the whole statement level. You can combine the propositions 'Roses are red' with 'Roses are thorny' into 'Roses are red AND roses are thorny' but you can't get to 'Roses are red and thorny' with only logic axioms. With predicate logic, we can.
is_red(roses) AND is_thorny(roses)
turns into
[is_red AND is_thorny](roses).
We get a new compound predicate, [is_red AND is_thorny], which means exactly what it says it means: a subject satisfies [is_red AND is_thorny] if and only if it both satisfies is_red and it satisfies is_thorny.

Naturally, you can use logical OR or any other logical operator as well.

• [is_even OR is_odd](4) //true - is_even(4) is true
  
• [is_even OR is_odd](3) //true - is_odd(3) is true
  
• [is_even OR is_odd](3.14159) //false - 3.14159 is neither even (evenly divisible by 2) nor odd (+/- 1 from an even number).

We can split it out the other way. Let's rewrite '3 is prime' as
is_prime(3).
Now we can also say is_prime(17), and it's obvious how 3 and 17 are connected: 3 and 17 are both in the group of subjects that satisfy is_prime:
if n is in { 3, 17 }, then is_prime(n).

We can even make predicates that take other predicates as subjects. Consider:

• is_true_for_at_least_one_flower(is_red) //true - 'rose' is a flower, and is_red(rose) is true.
  
• is_true_for_at_least_one_flower([is_even OR is_odd]) //false - flowers aren't numbers and therefore aren't in the even/odd dichotomy.

When are predicates equal to each other? Can I say that [is_red AND is_thorny] is the same as [is_thorny AND is_red]? They're obviously the same statement - logical AND doesn't care which order its arguments are in. So of course I can say it. Here's what it looks like in predicate logic:

• is_the_same_as_"[is_thorny AND is_red]"([is_thorny AND is_red]) //true - duh
  
• is_the_same_as_"[is_thorny AND is_red]"([is_red AND is_thorny]) //true - A AND B is the same as B AND A
  
• is_the_same_as_"[is_thorny AND is_red]"(roses) //false - roses are not a predicate
  
• is_the_same_as_"[is_thorny AND is_red]"(is_red) //false - fire engines are red but not thorny. is_red(fire engine) is true, but [is_thorny AND is_red](fire engine) is false. is_red answers differently from [is_thorny AND is_red] on the subject of fire engines, so they are different predicates.
  
• is_the_same_as_"[is_thorny AND is_red]"(is_the_same_as_"[is_thorny AND is_red]") //false - can you find a thing that [is_thorny AND is_red] and is_the_same_as_"[is_thorny AND is_red]" disagree on to prove it?

Two predicates are the same predicate if they agree on all subjects. A and B are the same predicate if, for all x, A(x) implies B(x) and B(x) implies A(x).

So what's all this got to do with set theory? Every set X immediately suggests a predicate: is_in_X. So for EGGS, { 'blue egg', 'red egg', 'yellow egg' }, we get the predicate is_in_EGGS.

• is_in_EGGS('blue egg') //true - 'blue egg' is in EGGS
  
• is_in_EGGS('red egg') //true - 'red egg' is in EGGS
  
• is_in_EGGS('yellow egg') //true - 'yellow egg' is in EGGS
  
• is_in_EGGS('roses') //false - roses is not in EGGS
  
• is_in_EGGS(EGGS) //false - EGGS is not an element of itself, as we explained in the previous long post.

Remember that sets don't care about number of repeats or order of elements. Neither do predicates. An object either satisfies the predicate or it does not. There is no 'satisfies the predicate four times'. predicate(thing) is either True or False, exactly the same way that 'thing' is either in a set or is not in a set. Likewise, there is no concept of order. It doesn't make sense to say that 'blue egg' satisfies is_in_EGGS before 'red egg' satisfies is_in_EGGS. They both satisfy the predicate; that's all there is to it. You can write the statement "is_in_EGGS('blue egg') AND is_in_EGGS('red egg')", and 'blue egg' is written before 'red egg' in that sentence, but that's just how you chose to write it and has nothing to do with is_in_EGGS itself. Similarly, { 'blue egg', 'red egg', 'yellow egg' } is exactly the same set as { 'red egg', 'yellow egg', 'blue egg' }. I for whatever reason chose to list the elements in a different order. That reflects on my personal writing style, not on the set EGGS itself.

There are a lot of parallels between the set EGGS and the predicate is_in_EGGS. In fact, it seems like the only difference is that EGGS is a set and is_in_EGGS is a predicate. Do we care about that? I maintain that EGGS and is_in_EGGS are one and the same entity. EGGS and is_in_EGGS both represent the same idea of "'blue egg', 'red egg', and 'yellow egg' are in/true, and everything else is out/false." Some people like to start from physical grocery carts and then remember all of the ways in which sets aren't like grocery carts. Personally, I think it's easier and less confusing to think of sets as being predicates.

If sets are actually predicates, and we can do logic to predicates, we should be able to do logic to sets. Let's define the set PIZZA_TOPPINGS to be { spinach, mushrooms, bacon }. Now we have the predicates (which are really sets) EGGS (a.k.a. is_in_EGGS) and PIZZA_TOPPINGS (a.k.a. is_in_PIZZA_TOPPINGS), plus our old predicate is_red (which doesn't have a corresponding set). Let's do some logic.

• EGGS('yellow egg') //true
  
• PIZZA_TOPPINGS)'yellow egg') //false - 'yellow egg' is not in PIZZA_TOPPINGS
  
• EGGS(EGGS) //false - EGGS is not in EGGS.
  
• [EGGS OR PIZZA_TOPPINGS](mushrooms) //true, because PIZZA_TOPPINGS](mushrooms) is true, and True OR anything is true.
  
• [EGGS AND PIZZA_TOPPINGS]('red egg') //false, because EGGS('red egg') is false, and False AND anything is false.
  
• [EGGS AND is_red]('red egg') //true - EGGS('red egg') is true and is_red('red egg') is true.
  
• [EGGS AND is_red]('blue egg') //false, because is_red('blue egg') is false.
  
• [is_red AND (EGGS OR PIZZA_TOPPINGS)](spinach) //false - is_red(spinach) is false.
  
• [is_red AND (EGGS OR PIZZA_TOPPINGS)](bacon) //true - is_red(bacon) is true, and [EGGS OR PIZZA_TOPPINGS](bacon) is true because PIZZA_TOPPINGS(bacon) is true.
  

Here's what those compound predicates look like as sets:

• [EGGS OR PIZZA_TOPPINGSS] = { 'blue egg', 'spinach', 'red egg', mushrooms, 'yellow egg', bacon }
  
• [EGGS AND PIZZA_TOPPINGSS] = { } //the empty set - nothing is in both EGGS and PIZZA_TOPPINGS
  
• [EGGS AND is_red] = { 'red egg' }
  
• [is_red AND (EGGS OR PIZZA_TOPPING)] = { 'red egg', bacon }
  

rong and I chatted a bit about the union of sets and the intersection of sets before. I gave him a definition, but it was really awkward and leads to lots of rote symbol pushing when applying it to complicated sets. Now I can give the easy definitions:
A ∪ B ("the union of A and B") = A ∨ B ("[A OR B]") = the set of things such that A(thing) OR B(thing)
A ∩ B ("the intersection of A and B") = A ∧ B ("[A AND B]") = the set of things such that A(thing) AND B(thing)

Now we can write the union and intersection of EGGS and PIZZA_TOPPINGS more easily:

EGGS ∪ PIZZA_TOPPINGS = { 'blue egg', 'spinach', 'red egg', mushrooms, 'yellow egg', bacon }

• EGGS ∩ PIZZA_TOPPINGS = { } //the empty set - nothing is in both EGGS and PIZZA_TOPPINGS

Can we construct our old set { EGGS, spinach, mushrooms, bacon } this way? Lets work backwards:

• X is in { EGGS, spinach, mushrooms, bacon } if X is the same as one of EGGS, spinach, mushrooms, or bacon.
  
• X is in { EGGS, spinach, mushrooms, bacon } if X is the same as EGGS or if PIZZA_TOPPINGS(X).
  
• X is in { EGGS, spinach, mushrooms, bacon } if is_the_same_as_"EGGS"(X) or if PIZZA_TOPPINGS(X).
  
• { EGGS, spinach, mushrooms, bacon } = [is_the_same_as_"EGGS" OR PIZZA_TOPPINGS]
  
• { EGGS, spinach, mushrooms, bacon } = { EGGS } ∪ PIZZA_TOPPINGS
  

{ EGGS } is the set that corresponds to the predicate is_the_same_as_"EGGS". EGGS is in { EGGS }, and is_the_same_as_"EGGS"(EGGS) is obviously true. EGGS is the only thing in { EGGS }, and if X satisfies is_the_same_as_"EGGS", then X is the same thing as EGGS, so EGGS is the only thing that satisfies is_the_same_as_"EGGS".

{ EGGS } is not the same thing as EGGS! If you understand which of the following statements are true and which are false, and can explain why, you have mastered this chapter.

• EGGS(EGGS)
  
• is_the_same_as_"EGGS"(EGGS)
  
• EGGS(is_the_same_as_"EGGS")
  
• is_the_same_as_"EGGS"(is_the_same_as_"EGGS")
  
• EGGS is an element of EGGS.
  
• EGGS is an element of { EGGS }.
  
• { EGGS } is an element of EGGS.
  
• { EGGS } is an element of { EGGS }.
  

You can substitute { 'blue egg', 'red egg', 'yellow egg' } in for EGGS if that helps you understand the questions better.

Next up: if all sets are also predicates, are all predicates also sets?

[Freeky]

is_the_same_as_"[is_thorny AND is_red]"(is_the_same_as_"[is_thorny AND is_red]") //false - can you find a thing that [is_thorny AND is_red] and is_the_same_as_"[is_thorny AND is_red]" disagree on to prove it?

I tried to read this and my brain cannot parse it. 

[Golden Applesauce]

is_the_same_as_"[is_thorny AND is_red]"(is_the_same_as_"[is_thorny AND is_red]") //false - can you find a thing that [is_thorny AND is_red] and is_the_same_as_"[is_thorny AND is_red]" disagree on to prove it?

I tried to read this and my brain cannot parse it. 

In retrospect, having to reread through five sets of quotes, parens, and brackets to double check that I got it right myself should have been a clue than that was too nested of an example. Will post a better example with clip art after a sandwich eventually.

[Nephew Twiddleton]

Ok so i realized today that in order not to set myself back significantly i should place out of precalculus over the summerand take calculus in september. I plan to start studying immediately. Where should i start?

[Golden Applesauce]

Crap. I'm not sure I actually remember what was in pre-calc vs. geometry before that and Calc I after that.

I think it was mostly algebra, but including solving equations that had lots of logarithms and exponents. Lots of functions, which were mostly combinations of add/subract/multiply/divide/log/exponents/sin/cosine, occasionally with stuff like "round up/down to the next number." Might have done a little stuff with polar coordinates and functions where both x and y depended on a third parameter. (Graphs like x = sin t, y = cos t, which gives you a circle for 0 <= t < 2 pi.

I think series, summations, and whether they converged or diverged, and how to tell, was a big deal. Or maybe that was part of actual calculus.

I'd make sure to know stuff like the logarithm identities and trigonometry identities like the back of your hand.

Play around with goofy functions in a programmable graphing environment (there are probably way better free online tools than the current gen of $100 graphing calculators.)

Do you know which teacher / which textbook you'll be using? If you have the syllabus for Calc, you can get the book early and skim the chapters that will be taught and see if there are any concepts in the early chapters you don't recognize.

[Mesozoic Mister Nigel]

Here, precalc is trigonometry.

[Don Coyote]

Crap. I'm not sure I actually remember what was in pre-calc vs. geometry before that and Calc I after that.

I think it was mostly algebra, but including solving equations that had lots of logarithms and exponents. Lots of functions, which were mostly combinations of add/subract/multiply/divide/log/exponents/sin/cosine, occasionally with stuff like "round up/down to the next number." Might have done a little stuff with polar coordinates and functions where both x and y depended on a third parameter. (Graphs like x = sin t, y = cos t, which gives you a circle for 0 <= t < 2 pi.

I think series, summations, and whether they converged or diverged, and how to tell, was a big deal. Or maybe that was part of actual calculus.

I'd make sure to know stuff like the logarithm identities and trigonometry identities like the back of your hand.

Play around with goofy functions in a programmable graphing environment (there are probably way better free online tools than the current gen of $100 graphing calculators.)

Do you know which teacher / which textbook you'll be using? If you have the syllabus for Calc, you can get the book early and skim the chapters that will be taught and see if there are any concepts in the early chapters you don't recognize.

Logarithms, trigonometry. In Precalc
Limits and derivatives in calculus 1
Summation, integrals in calculus 2

And don't both getting a graphing calculator. Get a cheap scientific one, like a TI-36xpro.
And know trig and log identities from memory will help as much as knowing how to to arithmetic, which is to say, a lot.

[Nephew Twiddleton]

Precalc is a combination of advanced algebra and trig. It's basically supposed to be, as is implied, a transition between algebra and calculus.

[Nephew Twiddleton]

In otherwords, I don't actually care what precal is, I just want to know where to start.

I got an A in college algebra (which at the time was apparently harder than the currently offered STEM algebra), so I can do this shit. I just need to know where algebra ends, and calculus begins.