Math with the passion of a THOUSAND SUNS

[Nephew Twiddleton]

Ok, so as mentioned in Open Bar, the math department head will sign off on my being able to take Precalculus in fall if I pass his final. I want to do this in case I fail the CLEP for Precalc. Now, he gave me a practice exam. I will be taking the actual exam one week from today. Would you math spags be willing to grade my practice test when I am done with it?

[Don Coyote]

Possibly.

[Nephew Twiddleton]

Possibly.

Cool. Will try to have done by Friday so I can scan it.

[GrannySmith]

Golden Applesauce, I really enjoyed your post on predicate logic but since I'm also a bit of a pervert when it comes to logic, I'll translate the entire post on predicate logic/set theory notation, which though it seems foreign at first, is for my brain easier to parse. The English language and indeed any human language I know (I don't know japanese!) sucks for this purpose. What I'll try to do is translate your post to a language I understand better - maybe some of you will sympathise. Do call me out if the result is competely unpedagogic/wrong.

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

PREDICATE LOGIC

[...]
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.

i will call is_red ≡ R (i use ≡ for "is defined as" or "is the name for", not to be confused with = which is "is equal to")
roses ≡ r
so I write R(r) for "roses are red".

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.

I started writing about predicates as functions that map to {True,False} but decided that I will just write "is true" or "is false" because talking about interpretations of truth values seems confusing to me at this point.

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).

I define:
AND ≡ ∧
is_thorny ≡ Th
so, with predicate logic, R(r)∧Th(r) turns into R∧Th(r).

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).

is_even ≡ E
is_odd ≡ O
OR ≡ ∨

• (E∨O)(4) is true because E(4) is true
  
• (E∨O)(3) is true because O(3) is true
  
• (E∨O)(3.14159) is false because E(3.14159) is false  (even ≡ divisible by 2) and O(3.14159) is false  (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).

is_prime ≡ P
here I want to introduce some set theoretic notation: ∈ ≡ "is in". So in set theory (not yet in predicate logic, see below) we write x∈X for "x is an element of the set X".
so, because P(3) is true and P(17) is true we have that
for every n∈{ 3, 17 }, P(n) is true.

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.

sorry this part i can't follow, please explain a little more!

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?

"is the same as" ≡ ↔

• (Th∧R ↔ Th∧R) is true  duh
  
• (Th∧R ↔ R∧Th)  is true because A AND B is the same as B AND A
  
• (Th∧R↔ r) is false because roses are not a predicate (here I would say that this is just meaningless)
  
• (Th∧R↔R) is false because fire engines are red but not thorny. With "fire engine"≡f_e we can say R(f_e)is true but Th∧R(f_e) is false. The predicate R( ) answers differently from the predicate Th∧R( ) on the subject f_e, so they are different.
  
• Th∧R↔(↔R) is false  (or again, meaningless; there are rules to combine propositions with the connectives ∧, ∨, ↔ etc.)
  

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).

"implies" ≡ →
So the predicate A↔B is true if the predicate ∀x( (A(x)→B(x)) ∧ (B(x)→A(x)) ) is true.

So what's all this got to do with set theory? Every set X immediately suggests a predicate: is_in_X.

As the quoted post suggests further down (the set is the same as the predicate), I will define X ≡ "is in X" and apply it to an object x as with all the predicates above: X( ).
In set theory we would write x∈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.

"blue egg" ≡ b
"red egg" ≡ r_e (remember r≡roses)
"yellow egg" ≡ y
so for E≡{ b,r_e,y } we get 

• E(b) is true  because b∈E
  
• E(r_e) is true because r_e∈E
  
• E(y) is true because y∈E
  
• E(r) is false because r∉E (∉ ≡ "not in")
  
• E(E) is false because in set theory a set can't be an element of itself.
  

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.

Here my notation is shitty - "E=E" doesn't express properly the boldfaced sentence above - the predicate is the set and the set is the predicate.

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.
  

spinach ≡ s
mushrooms ≡ m
bacon ≡ b_a
P_t ≡ { s, m, b_a}.
We have the predicates E, P_t, R.

• E(y) is true
  
• P_t(y) is false because y∉P_t
  
• E(E) is false because E∉E
  
• E∨P_t(m) is true, because P_t(m) is true
  
• E∧P_t(r_e) is false, because E(r_e) is false
  
• E∧R(r_e) is true because E(r_e) is true and R(r_e) is true.
  
• E∧R(b) is false, because R(b) is false.
  
• ( R∧(E∨P_t) )(s) is false because R(s) is false.
  
• ( R∧(E∨P_t) )(b_a) is true because R(b_a) is true, and (E∨P_t)(b_a) is true because P_t(b_a) 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 }
  

• E∨P_t = { b, s, r_e, m, y, b_a }
  
• E∧P_t = { } ≡  ∅
  
• E∧R = { r_e }
  
• R∧(E∨P_t) = { r_e, b_a }
  

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

• E∪P_t= { b,s,r_e,m,y,b_a}
  
• E∩P_t= ∅
  

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
  

• X∈{ E, s, m, b_a} if X=E or X=s or X=m or X=b_a
  
• X∈{ E, s, m, b_a} if X=E or P_t(X)
  
• X∈{ E, s, m, b_a} if E(X) or if P_t(X)
  
• { E, s, m, b_a } = E∨P_t
  
• { E, s, m, b_a}= { E }∪P_t
  

{ 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".

Hmm, here I'm running into trouble - I must have understood the "is_the_same_as_"EGGS" predicate wrong..?? My translation returns this:

{ E } is the set that corresponds to the predicate ↔E (but this is not making sense!). E∈{ E }, and E↔E is obviously true.
(should the predicate "is_the_same_as_"EGGS" be something like "(x↔E)(x)"?

{ 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 }.
  

{ E } ≠ E

• E(E)
  
• E↔E
  
• E(↔E)
  
• (↔E)↔E
  
• E∈E
  
• E∈{ E }
  
• { E }∈E
  
• { E }∈{ E }
  

I have the feeling I really fucked up the translation of "is_the_same_as_"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?

[GrannySmith]

hirley0, I REALLY want to understand your language. This is another attempt.

I believe there is no actual proof that 1+1=2, but rather, 1+1=2 is actually a definition.  Discuss?
i think i knew that character once upon a time | Set theory ? if set has only 0&1 | 2 is non¢ents

you disagree with the definition of 2≡{0,1} or logic should have more than 2 truth values? After too many years with classical logic I think logic should have more than two truth values.

[...]
^* of course we should have started from predicate logic, languages and theories, defined what a variable and a quantifier '(for all)' is, defined what a formula is, defined the rules that we make deductions with, defined what a proof is, defined substitution to formulas, and defined what a function is, so including all that the proof would be much longer!!
[...]
in time i will REMove much of your ARgument here | explain Y this REMinds me of a gree BUICK [Please do!!!]

RED _

Of course, part of my thought process is that I feel that it's better to spend a little more money and a little more time and be overprepared, and get straight A's, than to be underprepared and stressed and end up with a lower grade.

I should have spent more time defining everything from the beginning? Tell me more!

the 60's Music is on ch 10.1 & 10.2 / Jefferson Star ship
true? i cant tell if its Sig or not / i doubt they well say / Just GROUP

It was the sixties when eveN i could attend U unrestrained
&2 in '69 counter the attempt at my demise

What's a N0p?

NOP / no-op / no operation is a one byte assembly code instruction that does nothing, found in every assembly language (although not necessarily the same byte in each language, which I think Hirley0 might be referring to?)

I'm lost :/

[Mesozoic Mister Nigel]

Of course, part of my thought process is that I feel that it's better to spend a little more money and a little more time and be overprepared, and get straight A's, than to be underprepared and stressed and end up with a lower grade.

I should have spent more time defining everything from the beginning? Tell me more!

I was talking about my philosophy in approaching school, and referring to my immediate previous post (relevant portion quoted):

I was in a big rush to get it done until I decided to double-major, but then with all the additional prerequisites that brought I realized that it would be no fun at all if I didn't just fucking chill out and roll with it.

Nothing to do with what you "should" do, just relating my experience as an older returning student to Twid because he might find it useful since he's in the same boat.

[GrannySmith]

Of course, part of my thought process is that I feel that it's better to spend a little more money and a little more time and be overprepared, and get straight A's, than to be underprepared and stressed and end up with a lower grade.

I should have spent more time defining everything from the beginning? Tell me more!

I was talking about my philosophy in approaching school, and referring to my immediate previous post (relevant portion quoted):

I was in a big rush to get it done until I decided to double-major, but then with all the additional prerequisites that brought I realized that it would be no fun at all if I didn't just fucking chill out and roll with it.

Nothing to do with what you "should" do, just relating my experience as an older returning student to Twid because he might find it useful since he's in the same boat.

sorry, that was meant for hirley0, trying to decode/understand his language i went for connecting the colours 

[Mesozoic Mister Nigel]

Of course, part of my thought process is that I feel that it's better to spend a little more money and a little more time and be overprepared, and get straight A's, than to be underprepared and stressed and end up with a lower grade.

I should have spent more time defining everything from the beginning? Tell me more!

I was talking about my philosophy in approaching school, and referring to my immediate previous post (relevant portion quoted):

I was in a big rush to get it done until I decided to double-major, but then with all the additional prerequisites that brought I realized that it would be no fun at all if I didn't just fucking chill out and roll with it.

Nothing to do with what you "should" do, just relating my experience as an older returning student to Twid because he might find it useful since he's in the same boat.

sorry, that was meant for hirley0, trying to decode/understand his language i went for connecting the colours 

Oh, gotcha!

You just learn his language organically, over time. IME.

[Nephew Twiddleton]

Of course, part of my thought process is that I feel that it's better to spend a little more money and a little more time and be overprepared, and get straight A's, than to be underprepared and stressed and end up with a lower grade.

I should have spent more time defining everything from the beginning? Tell me more!

I was talking about my philosophy in approaching school, and referring to my immediate previous post (relevant portion quoted):

I was in a big rush to get it done until I decided to double-major, but then with all the additional prerequisites that brought I realized that it would be no fun at all if I didn't just fucking chill out and roll with it.

Nothing to do with what you "should" do, just relating my experience as an older returning student to Twid because he might find it useful since he's in the same boat.

sorry, that was meant for hirley0, trying to decode/understand his language i went for connecting the colours 

Oh, gotcha!

You just learn his language organically, over time. IME.

Yeah, pretty much. I still have no idea what he's talking about sometimes, but that means that it this point I do sometimes understand him.

[GrannySmith]

Ok, so as mentioned in Open Bar, the math department head will sign off on my being able to take Precalculus in fall if I pass his final. I want to do this in case I fail the CLEP for Precalc. Now, he gave me a practice exam. I will be taking the actual exam one week from today. Would you math spags be willing to grade my practice test when I am done with it?

hm, somehow i missed this before... definitely, no prob to grade the test!

[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.

Turns out I suck at responding to things. 

Take a menu. It lists a bunch of foods that a given restaurant can serve you. A menu is a kind of a set of foods; a dish is on the menu if it is in the set of dishes that are served by the restaurant. It doesn't matter if the printed menu has a given item listed more than once or in what order; you can either order the food or not.

Here are some menus:
Nguyen's Wonderful Wines has on it the proprietor's twelve favorite vintages.
Beth's Burger Bar has coke, fries, and a triple burger with large pickle.
Shalom's Salmon Shack has clams, pork cutlet, and milk.

We can write Nguyen's_Wonderful_Wines("2047 Pinot") = True if 2047 Pinot is in Nguyen's set of twelve favorite wines. That's just obnoxious mathematical notation, and the only real value it has is to remind us that we're doing math and not syllogisms. You can read NWW("2047 Pinot") as "2047 Pinot is served at Nguyen's Wonderful Wines." if you like (and I recommend you do if you're not comfortable with the math notation.) The equivalent expression in set notation is "2047 Pinot is an element of Nguyen's Wonderful Wines."

Importantly, none of the restaurants actually serve their own menu. You can read it, order from it, but you can't order the menu itself. The menu is not the meal.

"Nguyen's Wonderful Wines" is not served at Nguyen's Wonderful Wines. They only serve wines, not menus.
NWW(NWW) = False.

Good so far?

Now, all three of these restaurants operate in the same strip mall. Next to them is a highly specialized print shop. It specializes in printing menus, and curiously, it manages to stay profitable while only printing menus for its neighboring restaurants:

Pablo's Paradoxical Printer serves (sells?) the three menus Nguyen's Wondrous Wines, Beth's Burger Bar, and Shalom's Salmon Shack.

So we can say:
"Nguyen's Wonderful Wines" can be gotten at Pablo's Paradoxical Printer.
PPP(NWW) = true

But of course, PPP(2047 Pinot) = False. Pablo's Paradoxical Printer only prints menus; he doesn't actually have a stock of wine.

Similarly, PPP(PPP) is also false. Pablo is not at this time in the business of printing and selling catalogs (menus) of his own merchandise. He's more than happy to sell you some of the restaurant menus, but doesn't sell the list of things he sells any more than Beth will sell you her menu between two greasy buns. You get a triple burger with pickle or nothing.

Does that help at all?

[Bu🤠ns]

Thank you for this thread!!!


I need to catch up.