[2/19/2006 3:12 PM] <qwertydawom> Every day, we use to 'categorize' things.
[2/19/2006 3:12 PM] <qwertydawom> And, mathematicians use a special word for these categories.
[2/19/2006 3:13 PM] <qwertydawom> They call them : sets.
[2/19/2006 3:13 PM] -->| riftor (riftor@bu-C0240CF7.plus.com) has joined #lecture
[2/19/2006 3:13 PM] <qwertydawom> To describe these sets, mathematicians use symbols.
[2/19/2006 3:13 PM] =-= Mode #lecture +o riftor by qwertydawom
[2/19/2006 3:14 PM] <qwertydawom> Now, let's define what a 'set' is.
[2/19/2006 3:15 PM] <qwertydawom> Simply, sets are 'collection of things'.
[2/19/2006 3:15 PM] <qwertydawom> Now, how do we describe a set?
[2/19/2006 3:15 PM] <qwertydawom> There are two ways :
[2/19/2006 3:15 PM] <qwertydawom> the first one is writing all the elements of the set;
[2/19/2006 3:16 PM] <qwertydawom> the second one is telling a property that describes the set.
[2/19/2006 3:16 PM] <qwertydawom> Let's use an example.
[2/19/2006 3:17 PM] <qwertydawom> Say you want to show the set of the integers less than eight.
[2/19/2006 3:17 PM] <qwertydawom> The 2 ways to describe this set would be :
[2/19/2006 3:17 PM] <qwertydawom> 1° : S = {1, 2, 3, 4}
[2/19/2006 3:18 PM] <qwertydawom> 2° : S = {n in N| 1 <= n <= 4}
[2/19/2006 3:19 PM] <qwertydawom> These two definitions are exactly the same thing.
[2/19/2006 3:19 PM] <mu-tiger> qwertydawom: 
[2/19/2006 3:19 PM] <mu-tiger> you said a set of integers less than 8 but you only included 1-4
[2/19/2006 3:20 PM] <qwertydawom> :D
[2/19/2006 3:20 PM] <qwertydawom> good job mu
[2/19/2006 3:20 PM] <mu-tiger> so, isn't that a subset?
[2/19/2006 3:20 PM] <qwertydawom> ah! I see you know some stuff ;)
[2/19/2006 3:20 PM] <mu-tiger> a little bit
[2/19/2006 3:20 PM] <qwertydawom> that's good at least one is paying attention!
[2/19/2006 3:20 PM] <mu-tiger> my kid's right beside me
[2/19/2006 3:20 PM] <qwertydawom> now, what would be the correction mu?
[2/19/2006 3:21 PM] <mu-tiger> S=(1, 2, 3, 4, 5, 6, 7)
[2/19/2006 3:21 PM] <mu-tiger> and
[2/19/2006 3:22 PM] <mu-tiger> S= (n in N| 1<=n<=7)
[2/19/2006 3:22 PM] <mu-tiger> but i don't get why it's written that way
[2/19/2006 3:22 PM] <qwertydawom> yes, mu, you're totally right! :)
[2/19/2006 3:22 PM] <qwertydawom> ok, now, I shall explain these writings.
[2/19/2006 3:23 PM] <qwertydawom> for the first one, I guess there's no real problem?
[2/19/2006 3:23 PM] <qwertydawom> we just use "S" for set
[2/19/2006 3:23 PM] <mu-tiger> right
[2/19/2006 3:24 PM] <qwertydawom> and between "{", we put all the elements of the set, separated with a comma (or a semi-colon).
[2/19/2006 3:24 PM] <riftor> can I just interject here and say that sets are not ordered
[2/19/2006 3:24 PM] <riftor> and can not have duplicate elements
[2/19/2006 3:24 PM] <riftor> so { 1, 1, 1 } is not a valid set
[2/19/2006 3:24 PM] <riftor> and {1, 2, 3} is equal to {3, 1, 2}
[2/19/2006 3:26 PM] <qwertydawom> Indeed.
[2/19/2006 3:26 PM] <qwertydawom> Now, is it ok for the first way?
[2/19/2006 3:26 PM] <Ch4r> yeah
[2/19/2006 3:26 PM] <qwertydawom> alright, the second way then
[2/19/2006 3:27 PM] <qwertydawom> let's analyze what's between the brackets
[2/19/2006 3:27 PM] <qwertydawom> first, one can see : n in N
[2/19/2006 3:27 PM] <qwertydawom> what's that?
[2/19/2006 3:27 PM] <qwertydawom> well, "n" is just here to represent any number belonging ("in") to "N".
[2/19/2006 3:28 PM] <Ch4r> N is the set?
[2/19/2006 3:28 PM] <qwertydawom> but, what's N?
[2/19/2006 3:28 PM] <Ch4r> N = natural numbers?
[2/19/2006 3:28 PM] <mu-tiger> Numbers less than 8
[2/19/2006 3:28 PM] <mu-tiger> oh
[2/19/2006 3:29 PM] <qwertydawom> yes, N = natural numbers
[2/19/2006 3:29 PM] <qwertydawom> N is the set defined like this :
[2/19/2006 3:29 PM] <qwertydawom> N = {1, 2, 3, ... }
[2/19/2006 3:30 PM] <mu-tiger> so n is?
[2/19/2006 3:30 PM] <qwertydawom> n is any number of this set
[2/19/2006 3:30 PM] <mu-tiger> ok
[2/19/2006 3:30 PM] <qwertydawom> but! with the second part, we add a condition on "n"
[2/19/2006 3:32 PM] <riftor> okay, sorry to interupt again
[2/19/2006 3:32 PM] <riftor> I'd like to propose that we say the set of natural numbers starts at 0 and continues
[2/19/2006 3:32 PM] <riftor> N = {0,1,2,3,.... }
[2/19/2006 3:32 PM] <riftor> as this provides us with what's called a "closure" property for arithmetic
[2/19/2006 3:33 PM] <riftor> a closure property is when you say that if you have some state of some type, and you apply an operation to it, you get a new state that is of the same type as the old one
[2/19/2006 3:33 PM] <riftor> and its very important for us to have something called an "identity" property for arithmetic as well
[2/19/2006 3:34 PM] <riftor> that says, when we perform an operation with two inputs, we get as the result one of those inputs unchanged
[2/19/2006 3:34 PM] <riftor> so in arithmetic
[2/19/2006 3:34 PM] <riftor> A+0=A
[2/19/2006 3:34 PM] <riftor> and
[2/19/2006 3:34 PM] <riftor> A*1 = A
[2/19/2006 3:34 PM] <riftor> so those are the identity properties of multiplication and addition
[2/19/2006 3:34 PM] <riftor> which are part of our "closure"
[2/19/2006 3:35 PM] <riftor> so I feel 0 is important to have in our set of natural numbers :)
[2/19/2006 3:35 PM] <riftor> sorry for that tangent!
[2/19/2006 3:35 PM] <riftor> so yes.. N = {0,1,2,3...}
[2/19/2006 3:35 PM] <riftor> qwerty?
[2/19/2006 3:35 PM] <qwertydawom> ok!
[2/19/2006 3:36 PM] <qwertydawom> so now, if we want to take "natural integers less than eight", we'd only need to write :
[2/19/2006 3:36 PM] <qwertydawom> S = {n in N| n < 8}
[2/19/2006 3:36 PM] <qwertydawom> this "|" means "such that"
[2/19/2006 3:37 PM] <qwertydawom> and "n < 8", like most of you may know it means "n less than eight"
[2/19/2006 3:37 PM] <qwertydawom> so, now, I think you got the 2nd writing.
[2/19/2006 3:37 PM] <qwertydawom> Is it alright?
[2/19/2006 3:37 PM] <Ch4r> yep
[2/19/2006 3:37 PM] <mu-tiger> aye
[2/19/2006 3:38 PM] <Ch4r> hmm
[2/19/2006 3:38 PM] <Ch4r> just to clarify something
[2/19/2006 3:38 PM] <Ch4r> if I understand correctly...
[2/19/2006 3:38 PM] <Ch4r>  S = {n in N| 1 <= n <= 8} is NOT the same as S = {n in N| n < 8}
[2/19/2006 3:38 PM] <Ch4r> right?
[2/19/2006 3:38 PM] <qwertydawom> yep
[2/19/2006 3:38 PM] <Ch4r> ok
[2/19/2006 3:38 PM] <qwertydawom> if we define N to be {0, 1, ... }
[2/19/2006 3:39 PM] <riftor> the first is {1,2,3,4,5,6,7,8} the other is {0,1,2,3,4,5,6,7}
[2/19/2006 3:39 PM] <Ch4r> ok
[2/19/2006 3:39 PM] <qwertydawom> but, before I go on
[2/19/2006 3:39 PM] <qwertydawom> riftor will explain you what "n < 8" is.
[2/19/2006 3:41 PM] <qwertydawom> after all, it's a logic lecture, and these symbols belong to what's called "predicate logic"
[2/19/2006 3:41 PM] <riftor> heh okay :)
[2/19/2006 3:41 PM] <riftor> so what we have after our | is called a predicate
[2/19/2006 3:42 PM] <riftor> predicate calculus is the next step up from propositional logi which we saw yesterday
[2/19/2006 3:43 PM] <riftor> in propositional logic we were dealing with operators such as and, or and not that dealt with manipulating truth values
[2/19/2006 3:44 PM] <riftor> in predicate logic we are still writing statements that will ultimately evaluate to "true" or "false" but we have a lot more operators to play with
[2/19/2006 3:45 PM] <riftor> the corner stone of predicate calculus/logic is the use of quanitification, where we can bind a variable and provide a formal way of how we are going to deal with it
[2/19/2006 3:45 PM] <riftor> i will now introduce two new operators called the "universal" operator and the "existential" operator
[2/19/2006 3:45 PM] <riftor> the universal operator is drawn as an upside A, but in this IRC format I will have to use an A to represent it
[2/19/2006 3:46 PM] <riftor> the existential quantifies is drawn as a back to front E (so a west fast E if you like), but for this lecture I will have to write it as just E
[2/19/2006 3:46 PM] <riftor> now these "quantifiers" appear at the start of predicate statements to tell us about the 'bindings' of particular variables
[2/19/2006 3:47 PM] <riftor> the universal quantifier can be thought of as saying "for all something" and the existential can be though of saying "there exists some..."
[2/19/2006 3:48 PM] <riftor> so we might say
[2/19/2006 3:48 PM] <riftor> Ax:2x>x
[2/19/2006 3:48 PM] <riftor> this statement says, for all x, 2x is greater than x
[2/19/2006 3:48 PM] <riftor> so we are quantifying x, to be true for all its values
[2/19/2006 3:49 PM] <riftor> in fact that statement needs a bit of extra stuff
[2/19/2006 3:49 PM] <riftor> because we haven't said where x comes from
[2/19/2006 3:49 PM] <riftor> earlier we were looking at sets and saying n in N
[2/19/2006 3:50 PM] <riftor> here we want to do a similar thing but we have to use a set membership symbol, which we haven't seen yet
[2/19/2006 3:51 PM] <riftor> and sadly this is going to get really confusing as it is a capital epsilon character that looks like an E, so we are going to have to use something else for the moment
[2/19/2006 3:52 PM] <riftor> so lets just say Ax in N : 2x>x
[2/19/2006 3:52 PM] <riftor> so, for all x in the set of natural numbers, 2x is greater than x
[2/19/2006 3:52 PM] <riftor> now for existential, we are saying "there exists some..."
[2/19/2006 3:52 PM] <riftor> so
[2/19/2006 3:52 PM] <riftor> for example:-
[2/19/2006 3:53 PM] <riftor> Ex in N : x+3>10
[2/19/2006 3:53 PM] <riftor> saying, there exists some x in N where x+3>10
[2/19/2006 3:53 PM] <riftor> now the statement Ax in N : x+3>10 is not true, as 12+3 is not greater than 10, and 12 is in the set of natural numbers
[2/19/2006 3:54 PM] <riftor> but when we are using the existential operator, we are saying there exists some x in the set for which this is true
[2/19/2006 3:54 PM] <riftor> that could be just one value, or it could be all of them
[2/19/2006 3:55 PM] <qwertydawom> typo : "2+3" is not greater than 10
[2/19/2006 3:55 PM] <qwertydawom> and "2" is in N
[2/19/2006 3:55 PM] <riftor> sorry yes
[2/19/2006 3:55 PM] <riftor> i got myself confused
[2/19/2006 3:56 PM] <riftor> Ax in N : x+3>10 is not true, as 2+3 is not greater than 10, and 2 is in the set of natural numbers
[2/19/2006 3:57 PM] <riftor> now this predicate  calculus is very useful for defining other operations
[2/19/2006 3:58 PM] -->| CdPirate (CdPirate@bu-3B3CB0EE.as1.wxd.wexford.eircom.net) has joined #lecture
[2/19/2006 3:59 PM] <riftor> and we will probably see it more later, but be aware about it, and the fact that when we are writing these set definitions that we are using a "predicate" of kinds
[2/19/2006 3:59 PM] <riftor> quick last example
[2/19/2006 3:59 PM] <riftor> we might restrict the operation of some function, or even define it in a way by saying
[2/19/2006 3:59 PM] -->| skiddieleet (skiddielee@bu-F06C0E33.satx.res.rr.com) has joined #lecture
[2/19/2006 3:59 PM] <riftor> Ax in N: f(x)=x+2
[2/19/2006 4:00 PM] <riftor> saying, for all x in the set of natural numbers, a function f applied to x is equal to x+2
[2/19/2006 4:00 PM] <riftor> of course f might take other values than that of the natural numbers
[2/19/2006 4:00 PM] -->| switch (switch@bu-CB0B9791.hsd1.ma.comcast.net) has joined #lecture
[2/19/2006 4:01 PM] <riftor> or we might say:-
[2/19/2006 4:01 PM] <riftor> Ax in N, Ey in N : f(x)=y and y=x+2
[2/19/2006 4:01 PM] <riftor> for all x in the set of natural numbers, there exists a y also in the set of natural numbers such that f(x) is equal to y, and y=x+2
[2/19/2006 4:01 PM] <riftor> se where we have thrown in "and" from propositional logic, which we can do :)
[2/19/2006 4:02 PM] <riftor> any questions?
[2/19/2006 4:02 PM] -->| immortal (immortal@regime.gov) has joined #lecture
[2/19/2006 4:02 PM] <mu-tiger> uhm
[2/19/2006 4:02 PM] |<-- CdPirate has left irc.binaryuniverse.net (Quit: Well a yar har har and a compact disk...)
[2/19/2006 4:02 PM] <mu-tiger> my son and i need it dumbed down a bit
[2/19/2006 4:02 PM] <riftor> okay
[2/19/2006 4:03 PM] <riftor> so in predicate calculus, we can have functions
[2/19/2006 4:03 PM] <mu-tiger> yes
[2/19/2006 4:03 PM] <riftor> denoted in by some lower case letter, then paraeters
[2/19/2006 4:03 PM] <riftor> so f(x) for example
[2/19/2006 4:03 PM] <riftor> is the function f, that takes a variable x
[2/19/2006 4:03 PM] <mu-tiger> ok
[2/19/2006 4:03 PM] <riftor> we can use this function with a predicate to describe some of its operation
[2/19/2006 4:03 PM] <riftor> e.g.
[2/19/2006 4:04 PM] <riftor> Ax in N : f(x)>10
[2/19/2006 4:04 PM] <riftor> no we might not know whats going on inside f
[2/19/2006 4:04 PM] <riftor> but we are saying that for all numbers within the set of natural numbers N, that the function is defined, and produces a result greater than 10
[2/19/2006 4:05 PM] <riftor> Ax in N, Ey in N : f(x)=y and y=x+2
[2/19/2006 4:05 PM] <riftor> in that statement we are introducing out "existential" quanitifier also 
[2/19/2006 4:05 PM] <riftor> so say that, in english, for all x in the set of natural numbers N, there exists some y in the set of natural numbers (N), such that f(x)=y, and y=x+2
[2/19/2006 4:06 PM] <qwertydawom> e.g. : if x = 2, then : y = 4
[2/19/2006 4:07 PM] <riftor> or another example
[2/19/2006 4:07 PM] <mu-tiger> question
[2/19/2006 4:07 PM] <riftor> Ex in { monkeys, cats, dogs, humans } : x has no tail
[2/19/2006 4:08 PM] <riftor> a little bit of a silly example, but we are saying, there exists an x in the set of {monkeys, cars, dogs, humans} such that x has no tail
[2/19/2006 4:08 PM] <mu-tiger> does that define the function as "+"?
[2/19/2006 4:08 PM] <riftor> so we are saying, at least one of the types of animals has no tail
[2/19/2006 4:09 PM] <riftor> the function is aribtrary, we are really only defining the function by its result for all x in the natural numbers
[2/19/2006 4:09 PM] <mu-tiger> but you said
[2/19/2006 4:09 PM] <mu-tiger>  Ax in N, Ey in N : f(x)=y and y=x+2
[2/19/2006 4:09 PM] <mu-tiger> oi
[2/19/2006 4:09 PM] <mu-tiger> i'm confused
[2/19/2006 4:10 PM] <qwertydawom> well, it simply means that :
[2/19/2006 4:10 PM] <qwertydawom> if we take a natural integer, then, we can find another integer that will be our first one... plus two! :)
[2/19/2006 4:11 PM] <qwertydawom> let's take : 1.
[2/19/2006 4:11 PM] <qwertydawom> can you find this "y"?
[2/19/2006 4:11 PM] <mu-tiger> 3?
[2/19/2006 4:11 PM] <qwertydawom> yep
[2/19/2006 4:11 PM] <mu-tiger> my son answered that ;p
[2/19/2006 4:11 PM] <riftor> good work
[2/19/2006 4:11 PM] <qwertydawom> and, you could do the same thing FOR ALL x.
[2/19/2006 4:11 PM] <mu-tiger> ok
[2/19/2006 4:12 PM] <mu-tiger> we got it now
[2/19/2006 4:12 PM] <mu-tiger> ty
[2/19/2006 4:12 PM] <riftor> cool
[2/19/2006 4:14 PM] <qwertydawom> so, let's get back to our first example
[2/19/2006 4:14 PM] <qwertydawom> the natural integers less than eight
[2/19/2006 4:14 PM] <qwertydawom> when I first wrote the set S, mu pointed out the mistake
[2/19/2006 4:15 PM] <qwertydawom> and! she employed the word "subset"
[2/19/2006 4:15 PM] <qwertydawom> indeed, if we define T to be :
[2/19/2006 4:15 PM] <qwertydawom> T = {1, 2, 3, 4}
[2/19/2006 4:15 PM] <qwertydawom> and S = {0, 1, 2, 3, 4, 5, 6, 7}
[2/19/2006 4:16 PM] <qwertydawom> we can see that all the elements of T are in S.
[2/19/2006 4:16 PM] <qwertydawom> that's the definition of a subset.
[2/19/2006 4:16 PM] <qwertydawom> "The set T is a subset of S if every element of T is also in S"
[2/19/2006 4:17 PM] <qwertydawom> or, with our dear logic symbols : x \in T ==> x \in T
[2/19/2006 4:18 PM] <qwertydawom> where '\in' is the epsilon riftor was talking about
[2/19/2006 4:18 PM] <mu-tiger> what is '\'?
[2/19/2006 4:18 PM] <mu-tiger> oh
[2/19/2006 4:18 PM] <mu-tiger> my bad
[2/19/2006 4:18 PM] <mu-tiger> which means, again?
[2/19/2006 4:18 PM] <qwertydawom> <qwertydawom> or, with our dear logic symbols : x \in T ==> x \in S
[2/19/2006 4:19 PM] <qwertydawom> well : x \in T means that the set T contains the element x
[2/19/2006 4:19 PM] <riftor> i'd just like to indicate that in predicate calculus we would say, Ax \in T: x \in S
[2/19/2006 4:20 PM] <qwertydawom> yep
[2/19/2006 4:20 PM] <qwertydawom> is that ok?
[2/19/2006 4:20 PM] <mu-tiger> uhm
[2/19/2006 4:21 PM] <mu-tiger> i don't understand what the function of '\' is
[2/19/2006 4:21 PM] <qwertydawom> forget about it if you want
[2/19/2006 4:21 PM] <mu-tiger> ok
[2/19/2006 4:21 PM] <mu-tiger> lol
[2/19/2006 4:21 PM] <qwertydawom> it's just that, since we cannot represent the 'epsilon', I use '\in'
[2/19/2006 4:21 PM] <mu-tiger> o
[2/19/2006 4:22 PM] <mu-tiger> ok i gotcha
[2/19/2006 4:22 PM] <qwertydawom> :)
[2/19/2006 4:22 PM] <qwertydawom> good
[2/19/2006 4:22 PM] <qwertydawom> so, now, let N be the set of the natural integers
[2/19/2006 4:22 PM] <qwertydawom> what can you say about S with respect to N?
[2/19/2006 4:22 PM] <mu-tiger> it's a subset <_<
[2/19/2006 4:23 PM] <qwertydawom> that's right! :)
[2/19/2006 4:25 PM] <qwertydawom> alright, so, let's recall our stars of the day... S and T!
[2/19/2006 4:26 PM] <qwertydawom> you've seen that the numbers '1, 2, 3 and 4' were in the two sets, right?
[2/19/2006 4:26 PM] <mu-tiger> yup
[2/19/2006 4:27 PM] |<-- Ignite has left irc.binaryuniverse.net (Quit: i'm to tired to take all this in, can't keep my eyes open, night all and keep it up qwertydawom good work)
[2/19/2006 4:27 PM] <qwertydawom> so, we would say that "the intersection of S and T" is the set containing "1, 2, 3 and 4" (in this case S inter. T would be T)
[2/19/2006 4:28 PM] <qwertydawom> Generally speaking, the intersection of two sets S and T consists of all elements present in both set.
[2/19/2006 4:28 PM] =-= Mode #lecture +v skiddieleet by qwertydawom
[2/19/2006 4:29 PM] <qwertydawom> We denote the intersection by a reversed "U". 
[2/19/2006 4:29 PM] <qwertydawom> But, here I'll just use "inter".
[2/19/2006 4:29 PM] <riftor> (upside down U, it looks like a big 'n')
[2/19/2006 4:30 PM] <qwertydawom> So : S inter T = {x| x \in S and x \in T}
[2/19/2006 4:31 PM] <qwertydawom> got it?
[2/19/2006 4:31 PM] <Ch4r> yea
[2/19/2006 4:31 PM] <qwertydawom> ok, we'll see it right now with a little practice exercise :
[2/19/2006 4:33 PM] <qwertydawom> Let A = {x \in N | x >= 5} and B = {x \in N | x < 9}.
[2/19/2006 4:33 PM] <qwertydawom> What would be 'A inter B'?
[2/19/2006 4:33 PM] <skiddieleet> A inter B = { 5, 6, 7, 8 }
[2/19/2006 4:33 PM] =-= Mode #lecture -m  by qwertydawom
[2/19/2006 4:33 PM] <qwertydawom> indeed :)
[2/19/2006 4:33 PM] <qwertydawom> do you all agree with it?
[2/19/2006 4:34 PM] <mu-tiger> uhm
[2/19/2006 4:34 PM] <Ch4r> yep
[2/19/2006 4:34 PM] <mu-tiger> i think so
[2/19/2006 4:34 PM] <qwertydawom> k.
[2/19/2006 4:34 PM] =-= Mode #lecture +m  by qwertydawom
[2/19/2006 4:34 PM] <riftor> A = {5,6,7,8,9...}
[2/19/2006 4:34 PM] <riftor> B = {0,1,2,3,4,5,6,7,8}
[2/19/2006 4:34 PM] <riftor> A inter B = {5,6,7,8}
[2/19/2006 4:35 PM] <qwertydawom> yes, and now, a little bit 'trickier' :
[2/19/2006 4:36 PM] <qwertydawom> Let A = {1, 5, 8} and B = {2, 7, 156}.
[2/19/2006 4:36 PM] <qwertydawom> What's 'A inter B'?
[2/19/2006 4:36 PM] =-= Mode #lecture -m  by qwertydawom
[2/19/2006 4:36 PM] <skiddieleet> empty set?
[2/19/2006 4:36 PM] <Ch4r> yeah
[2/19/2006 4:36 PM] <qwertydawom> indeed :)
[2/19/2006 4:36 PM] <mu-tiger> :)
[2/19/2006 4:36 PM] <skiddieleet> I win
[2/19/2006 4:36 PM] <skiddieleet> :P
[2/19/2006 4:37 PM] <qwertydawom> So, if you know what the empty set is, then you might also know the symbol? :)
[2/19/2006 4:37 PM] =-= Mode #lecture +m  by qwertydawom
[2/19/2006 4:37 PM] <Ch4r> it's like a 0 with a line through it, right?
[2/19/2006 4:37 PM] <qwertydawom> right ;)
[2/19/2006 4:37 PM] <--| immortal has left #lecture
[2/19/2006 4:38 PM] <qwertydawom> So, when the intersection of two sets is empty, we say that these two sets are "disjoint" or "mutually exclusive".
[2/19/2006 4:38 PM] <qwertydawom> And, now, who makes a bright remark and notice that the intersection really looks like an operator we've seen just yesterday? :)
[2/19/2006 4:38 PM] =-= Mode #lecture -m  by qwertydawom
[2/19/2006 4:39 PM] <skiddieleet> Ch4r?
[2/19/2006 4:40 PM] <Ch4r> skiddieleet?
[2/19/2006 4:40 PM] <mu-tiger> !=
[2/19/2006 4:40 PM] <mu-tiger> ?
[2/19/2006 4:40 PM] <skiddieleet> 
[2/19/2006 4:40 PM] <mu-tiger> i dunno
[2/19/2006 4:40 PM] <skiddieleet> that was my answer to the question :P
[2/19/2006 4:40 PM] |<-- switch has left irc.binaryuniverse.net (Quit: leaving)
[2/19/2006 4:41 PM] <qwertydawom> yes :)
[2/19/2006 4:41 PM] <qwertydawom> so, complete this sentence :
[2/19/2006 4:41 PM] <qwertydawom> when we take 'A inter B', we want the elements of A ... the elements of B.
[2/19/2006 4:42 PM] <mu-tiger> to contain?
[2/19/2006 4:42 PM] <skiddieleet> that are also?
[2/19/2006 4:42 PM] <qwertydawom> a single word :)
[2/19/2006 4:42 PM] <qwertydawom> a logic operator ;)
[2/19/2006 4:43 PM] <skiddieleet> inclusive?
[2/19/2006 4:43 PM] <Ch4r> in?
[2/19/2006 4:43 PM] <qwertydawom> no no :)
[2/19/2006 4:43 PM] <qwertydawom> we've seen it yesterday
[2/19/2006 4:43 PM] <Ch4r> or? ;/
[2/19/2006 4:43 PM] <Ch4r> no
[2/19/2006 4:43 PM] <Ch4r> lol
[2/19/2006 4:43 PM] <skiddieleet> I missed class yesterday
[2/19/2006 4:43 PM] <qwertydawom> mmk
[2/19/2006 4:44 PM] <qwertydawom> well, the answer was... "AND" :)
[2/19/2006 4:44 PM] <mu-tiger> lol
[2/19/2006 4:44 PM] <skiddieleet> ohhhh
[2/19/2006 4:44 PM] <mu-tiger> nice to know we don't disappoint our teachers <_<
[2/19/2006 4:44 PM] <skiddieleet> I saw that in the notes someone gave me from yesterday
[2/19/2006 4:45 PM] <qwertydawom> hehe
[2/19/2006 4:45 PM] <qwertydawom> and now, seeing that Ch4r wants to place it, we'll see what is called the "union" :)
[2/19/2006 4:45 PM] <Ch4r> hmm... I thoought about 'and', but wouldn't that mean it would be all the elements from A as well as the elements of B, which would be incorrect, s ince we're talking about intersections? -_-
[2/19/2006 4:45 PM] <Ch4r> haha
[2/19/2006 4:46 PM] <qwertydawom> Well, it's correct, since we want the elements that belong to A AND to B ;)
[2/19/2006 4:46 PM] <skiddieleet> the operation AND means they have to be in both
[2/19/2006 4:46 PM] <Ch4r> ok right
[2/19/2006 4:47 PM] <qwertydawom> yes, and now, we'll see the union is like "or" ;)
[2/19/2006 4:48 PM] <qwertydawom> The union of two sets S and T, denoted by "S U T", is composed by the elements that are in A or those that are in B.
[2/19/2006 4:48 PM] <qwertydawom> i.e. : S U T = {x | x \in S OR x \in T}
[2/19/2006 4:49 PM] <qwertydawom> So, little exercise :
[2/19/2006 4:49 PM] <qwertydawom> A = {1, 2, 4}
[2/19/2006 4:49 PM] <qwertydawom> B = {3, 5}
[2/19/2006 4:49 PM] <qwertydawom> What is "A U B"?
[2/19/2006 4:50 PM] <skiddieleet> A U B = { 1, 2, 3, 4, 5 }
[2/19/2006 4:50 PM] <mu-tiger> {1, 2, 3, 4, 5}?
[2/19/2006 4:50 PM] <mu-tiger> uh
[2/19/2006 4:50 PM] <mu-tiger> >:(
[2/19/2006 4:50 PM] <skiddieleet> sorry
[2/19/2006 4:50 PM] <mu-tiger> lol
[2/19/2006 4:50 PM] <mu-tiger> np
[2/19/2006 4:50 PM] <qwertydawom> yes, you're both right :)
[2/19/2006 4:51 PM] <qwertydawom> and, now, let's see if you're smart cookies ;)
[2/19/2006 4:51 PM] <qwertydawom> A = {1, 3, 5, 7, .... }
[2/19/2006 4:51 PM] <skiddieleet> I love cookies
[2/19/2006 4:51 PM] <qwertydawom> B = {2, 4, 6, 8, ... }
[2/19/2006 4:51 PM] <qwertydawom> what'd be "A U B"?
[2/19/2006 4:51 PM] <Ch4r> A U B = {1,2,3,4,5,6,7,8...}
[2/19/2006 4:51 PM] <skiddieleet> A U B = { 1, 2, 3, ...}
[2/19/2006 4:51 PM] <Ch4r> ,...*
[2/19/2006 4:51 PM] <Ch4r> yea
[2/19/2006 4:51 PM] <qwertydawom> which is? :)
[2/19/2006 4:51 PM] <skiddieleet> N
[2/19/2006 4:52 PM] <skiddieleet> all Natural numbers
[2/19/2006 4:52 PM] <qwertydawom> indeed! :)
[2/19/2006 4:52 PM] <mu-tiger> :)
[2/19/2006 4:52 PM] <qwertydawom> well, we've previously defined N to contain "0", but... that's the idea
[2/19/2006 4:52 PM] <skiddieleet> I didn't know that
[2/19/2006 4:52 PM] <qwertydawom> in fact, this set would be denoted "N*", the star showing that "0" isn't in it.
[2/19/2006 4:53 PM] <skiddieleet> is that our N, or the Natural Numbers have 0 also?
[2/19/2006 4:53 PM] <mu-tiger> natural #s have 0
[2/19/2006 4:53 PM] <skiddieleet> let's do relations :P
[2/19/2006 4:53 PM] <Ch4r> uhm
[2/19/2006 4:54 PM] <qwertydawom> well, to be honest, it varies with the countries.
[2/19/2006 4:54 PM] <Ch4r> from my understanding, N contains 0 when used in set theory and computer science
[2/19/2006 4:54 PM] <Ch4r> and it doesn't every where else
[2/19/2006 4:54 PM] <qwertydawom> well, I know that in US you're taught : N = {1, 2, 3, ...}
[2/19/2006 4:54 PM] <qwertydawom> but, for example in France, we're told that : N = {0, 1, 2, ... }
[2/19/2006 4:55 PM] <skiddieleet> Z+
[2/19/2006 4:55 PM] <mu-tiger> i was taught 0
[2/19/2006 4:55 PM] <qwertydawom> but, like riftor previoulsy stated, it's better to include "0" in our N.
[2/19/2006 4:55 PM] <skiddieleet> in my discrete math book N is 0, 1...
[2/19/2006 4:55 PM] <qwertydawom> ok then ;)
[2/19/2006 4:55 PM] <skiddieleet> Z is the set of all Integers, and Z+ is 1, 2, ...
[2/19/2006 4:55 PM] <skiddieleet> positive integers
[2/19/2006 4:55 PM] <qwertydawom> yes
[2/19/2006 4:56 PM] <qwertydawom> so, now, let's do some 'counting' :)
[2/19/2006 4:56 PM] <qwertydawom> Let's take a set with three elements :
[2/19/2006 4:56 PM] <qwertydawom> A = {a, b, c}
[2/19/2006 4:56 PM] <qwertydawom> Who can show me all the subsets of A?
[2/19/2006 4:57 PM] <mu-tiger> {a}, {a, b}?
[2/19/2006 4:57 PM] <mu-tiger> erm
[2/19/2006 4:57 PM] <mu-tiger> wait
[2/19/2006 4:57 PM] <Ch4r> {a}
[2/19/2006 4:57 PM] <Ch4r> {b}
[2/19/2006 4:57 PM] <Ch4r> {c}
[2/19/2006 4:57 PM] <Ch4r> {a,b}
[2/19/2006 4:57 PM] <mu-tiger> {ac}
[2/19/2006 4:57 PM] <Ch4r> {a,c}
[2/19/2006 4:57 PM] <Ch4r> {b,c}
[2/19/2006 4:57 PM] <Ch4r> right?
[2/19/2006 4:57 PM] <qwertydawom> hehe, you forgot two in fact :)
[2/19/2006 4:57 PM] <Ch4r> =x
[2/19/2006 4:58 PM] <qwertydawom> but, I shall explain your mistake (which is normal at that point) :
[2/19/2006 4:58 PM] <qwertydawom> 1) "every set is a subset of itself"
[2/19/2006 4:58 PM] <qwertydawom> so, can you add one? :)
[2/19/2006 4:58 PM] <Ch4r> ah
[2/19/2006 4:58 PM] <mu-tiger> o.0
[2/19/2006 4:59 PM] <skiddieleet> is the empty set a subset?
[2/19/2006 4:59 PM] <qwertydawom> that is "2)" :)
[2/19/2006 4:59 PM] <skiddieleet> :P
[2/19/2006 4:59 PM] <qwertydawom> 2) "the empty set is a subset of every set"
[2/19/2006 4:59 PM] <qwertydawom> so, now, who can finish the list? :)
[2/19/2006 4:59 PM] <mu-tiger> huh?
[2/19/2006 4:59 PM] <Ch4r> {a,b,c}
[2/19/2006 4:59 PM] <Ch4r> and 0 with a line through it
[2/19/2006 4:59 PM] <Ch4r> :P
[2/19/2006 4:59 PM] <skiddieleet> empty set, {a}, {a, b}, {a, b, c}
[2/19/2006 4:59 PM] <qwertydawom> yes, ch4r's got them all ;)
[2/19/2006 5:00 PM] <mu-tiger> :)
[2/19/2006 5:00 PM] <mu-tiger> smarties
[2/19/2006 5:00 PM] <skiddieleet> oh
[2/19/2006 5:00 PM] <qwertydawom> how many do we have ch4r?
[2/19/2006 5:00 PM] <skiddieleet> 8?
[2/19/2006 5:00 PM] <Ch4r> 8 
[2/19/2006 5:00 PM] <qwertydawom> yep :)
[2/19/2006 5:00 PM] <Ch4r> ;D
[2/19/2006 5:00 PM] <skiddieleet> 2^n
[2/19/2006 5:00 PM] <qwertydawom> grr :p
[2/19/2006 5:00 PM] <qwertydawom> now, mu, just do the same thing with :
[2/19/2006 5:00 PM] <qwertydawom> B = {a, b}
[2/19/2006 5:01 PM] <mu-tiger> k
[2/19/2006 5:01 PM] <mu-tiger> {a}, {b}, empty set, and {ab}
[2/19/2006 5:01 PM] <qwertydawom> alright
[2/19/2006 5:01 PM] <qwertydawom> how many?
[2/19/2006 5:01 PM] <mu-tiger> 4
[2/19/2006 5:01 PM] <qwertydawom> true :)
[2/19/2006 5:01 PM] <qwertydawom> so, you see that :
[2/19/2006 5:02 PM] <qwertydawom> for a set with "2" elements, you have "4" subsets
[2/19/2006 5:02 PM] <qwertydawom> for a set with "3" elements, you have "8" subsets.
[2/19/2006 5:02 PM] <qwertydawom> and... skiddieleet will continue for me ;)
[2/19/2006 5:03 PM] <skiddieleet> so a set with n elements has 2^n subsets
[2/19/2006 5:03 PM] <qwertydawom> that's it :)
[2/19/2006 5:03 PM] <qwertydawom> does everyone get it?
[2/19/2006 5:03 PM] <Ch4r> yep
[2/19/2006 5:03 PM] <mu-tiger> nope
[2/19/2006 5:04 PM] <mu-tiger> 2^n?
[2/19/2006 5:04 PM] <Ch4r> 2 to the nth power
[2/19/2006 5:04 PM] <qwertydawom> yep
[2/19/2006 5:04 PM] <qwertydawom> 2^2 = 4
[2/19/2006 5:04 PM] <mu-tiger> i understand that
[2/19/2006 5:04 PM] <mu-tiger> but
[2/19/2006 5:04 PM] <qwertydawom> 2^3 = 8
[2/19/2006 5:04 PM] <mu-tiger> ok
[2/19/2006 5:04 PM] <mu-tiger> ty
[2/19/2006 5:04 PM] <mu-tiger> got it
[2/19/2006 5:04 PM] <qwertydawom> np
[2/19/2006 5:04 PM] <qwertydawom> try on a paper with a set containing 4 elements ;)
[2/19/2006 5:05 PM] <qwertydawom> (I wouldn't advice you to try with 10 elements, but... :P)
[2/19/2006 5:05 PM] <qwertydawom> Ok, so, we say that "the set of all the subsets of S is called the power set of S".
[2/19/2006 5:06 PM] <qwertydawom> It is denoted by : 2^S
[2/19/2006 5:06 PM] <qwertydawom> The power set is a set.. of sets! :)
[2/19/2006 5:06 PM] <Ch4r> trickeh
[2/19/2006 5:06 PM] <qwertydawom> so, with a two elements set, the power set is like :
[2/19/2006 5:06 PM] <mu-tiger> uhm
[2/19/2006 5:07 PM] <mu-tiger> i gotta question
[2/19/2006 5:07 PM] <qwertydawom> 2^S = {S_1, S_2, S_3, S_4}
[2/19/2006 5:07 PM] <qwertydawom> go on mu :)
[2/19/2006 5:07 PM] <mu-tiger> ty, sry i interrupted but here:
[2/19/2006 5:07 PM] <mu-tiger> 15:02 <@qwertydawom> try on a paper with a set containing 4 elements ;)
[2/19/2006 5:07 PM] <mu-tiger> if i went A= {a,b,c,d}
[2/19/2006 5:07 PM] <mu-tiger> and one subset was like
[2/19/2006 5:08 PM] <mu-tiger> {a,b}
[2/19/2006 5:08 PM] <mu-tiger> would i have to write another subset as {b,a}?
[2/19/2006 5:08 PM] <qwertydawom> no! :)
[2/19/2006 5:08 PM] <mu-tiger> ok
[2/19/2006 5:08 PM] <mu-tiger> i didn't think so but wasn't sure
[2/19/2006 5:08 PM] <mu-tiger> ty
[2/19/2006 5:08 PM] <qwertydawom> like riftor mentioned at the beginning :
[2/19/2006 5:08 PM] <mu-tiger> ok
[2/19/2006 5:08 PM] <qwertydawom> {a, b} is the same set as {b, a}
[2/19/2006 5:08 PM] <mu-tiger> i thought so
[2/19/2006 5:08 PM] <qwertydawom> ok ;)
[2/19/2006 5:08 PM] <mu-tiger> just ordered differently
[2/19/2006 5:09 PM] <mu-tiger> ty, go on, pls
[2/19/2006 5:09 PM] <qwertydawom> Well, in fact, with power sets, our lecture will end.. :)
[2/19/2006 5:09 PM] <Ch4r> awesome lecture qwerty :D
[2/19/2006 5:09 PM] <mu-tiger> woohoo!
[2/19/2006 5:09 PM] <Ch4r> and riftor if you're still here
[2/19/2006 5:09 PM] <qwertydawom> And, tomorrow, we'll see how to 'picture' relations between sets ;)