[2/25/2006 3:15 PM] k [2/25/2006 3:15 PM] just one question though [2/25/2006 3:16 PM] do you all know what Venn diagrams are? [2/25/2006 3:16 PM] yes [2/25/2006 3:18 PM] others know it too? [2/25/2006 3:19 PM] yes [2/25/2006 3:20 PM] alright, for those who don't answer and don't know what it is : http://www.sdcoe.k12.ca.us/score/actbank/tvenn.htm [2/25/2006 3:20 PM] now, I can begin :) [2/25/2006 3:21 PM] So, in our last lectures, we studied the 'union' and 'interesction' of sets. [2/25/2006 3:21 PM] And, there are rules that apply to these operators. [2/25/2006 3:22 PM] If you use Venn diagrams, it should be very easy to check these rules. :) [2/25/2006 3:22 PM] There are three main properties : [2/25/2006 3:22 PM] 1° Commutivity : [2/25/2006 3:22 PM] A inter B = B inter A [2/25/2006 3:22 PM] A U B = B U A [2/25/2006 3:23 PM] is it ok? [2/25/2006 3:23 PM] <--| Syr0kill has left #lecture (Leaving) [2/25/2006 3:24 PM] -->| grimR (grim@bu-486EF152.hsd1.md.comcast.net) has joined #lecture [2/25/2006 3:24 PM] yea.. [2/25/2006 3:24 PM] alright, so, 2nd property : [2/25/2006 3:24 PM] 2° Associativity : [2/25/2006 3:25 PM] (A inter B) inter C = A inter (B inter C) [2/25/2006 3:25 PM] (A U B) U C = A U (B U C) [2/25/2006 3:25 PM] ok? [2/25/2006 3:25 PM] * Ch4r nods [2/25/2006 3:25 PM] yep [2/25/2006 3:25 PM] k [2/25/2006 3:26 PM] so, 3rd property : [2/25/2006 3:26 PM] 3° Distributivity : [2/25/2006 3:26 PM] A inter (B U C) = (A inter B) U (A inter C) [2/25/2006 3:26 PM] A U (B inter C) = (A U B) inter (A U C) [2/25/2006 3:27 PM] alright? [2/25/2006 3:27 PM] mhm [2/25/2006 3:27 PM] you can see it if you use Venn diagrams ;) [2/25/2006 3:28 PM] tell me when it's ok [2/25/2006 3:29 PM] it's ok :p [2/25/2006 3:29 PM] ok ;) [2/25/2006 3:29 PM] there's another property. I forgot =| [2/25/2006 3:30 PM] -->| Narada (Node1@bu-869AAE49.client.mchsi.com) has joined #lecture [2/25/2006 3:30 PM] Sorry I'm late. [2/25/2006 3:30 PM] np, just stfu and learn ;) [2/25/2006 3:30 PM] hehe [2/25/2006 3:31 PM] the long version of npjstfual [2/25/2006 3:31 PM] ok, so, now we've seen those three rules [2/25/2006 3:32 PM] who makes the connection with algebra and show me algebra operators for which these rules apply? :) [2/25/2006 3:32 PM] hrmmm [2/25/2006 3:32 PM] *looks on the calculator keys* [2/25/2006 3:32 PM] commutivity: a*b = b*a, right? [2/25/2006 3:32 PM] yep [2/25/2006 3:33 PM] w00t [2/25/2006 3:33 PM] ;P [2/25/2006 3:33 PM] distributive = a(bc) ab+ac [2/25/2006 3:33 PM] no [2/25/2006 3:33 PM] a(b+c) ab+ac [2/25/2006 3:33 PM] yes :) [2/25/2006 3:33 PM] and associative? [2/25/2006 3:33 PM] associativity: a*b*c = a*c*b [2/25/2006 3:34 PM] abc = acb = cab = bac [2/25/2006 3:35 PM] you need parentheses :) [2/25/2006 3:35 PM] what you show here is just commutativity [2/25/2006 3:35 PM] a(bc)=b(ac) [2/25/2006 3:36 PM] and.. Narada is... right! :) [2/25/2006 3:36 PM] another one : a+(b+c) = (a+b)+c :) [2/25/2006 3:37 PM] Ok, so now, you remember that we've seen the TRUTH tables, right? [2/25/2006 3:37 PM] maybe [2/25/2006 3:37 PM] =-= Mode #lecture +m by qwertydawom [2/25/2006 3:37 PM] =-= Mode #lecture +v Ch4r by qwertydawom [2/25/2006 3:37 PM] =-= Mode #lecture +v Elmo_ by qwertydawom [2/25/2006 3:37 PM] =-= Mode #lecture +v Narada by qwertydawom [2/25/2006 3:38 PM] So, we are now going to explain the meaning of 'truth' in mathematics. [2/25/2006 3:38 PM] Definition : Something is said to be true if there are NO exceptions. [2/25/2006 3:39 PM] In life, when we say something is true, we take it as 'most of the times it's true'. [2/25/2006 3:40 PM] For example, when we say that Linux owns Windows, it doesn't mean that every single distro owns it, but just that most do! :) [2/25/2006 3:40 PM] -sorry for having chosen this.. oh so controversial subject- :P [2/25/2006 3:40 PM] * Ch4r glares [2/25/2006 3:40 PM] ;) [2/25/2006 3:40 PM] ./lecture -flame.war [2/25/2006 3:40 PM] I thought it was funny... [2/25/2006 3:41 PM] :) [2/25/2006 3:41 PM] But, for mathematicians, it isn't the same thing, if it's true, then there should be NO exceptions. [2/25/2006 3:41 PM] Let's take an example. [2/25/2006 3:42 PM] like Suse or Ubuntu :-) [2/25/2006 3:42 PM] The inverse (or 'reciprocal') of a number is : one divided by this number. [2/25/2006 3:42 PM] e.g. : the inverse of 5 is : 1/5 [2/25/2006 3:42 PM] Now, is the following sentence true? : [2/25/2006 3:43 PM] "Every number has an inverse" [2/25/2006 3:43 PM] Yes. [2/25/2006 3:43 PM] You fell in the trap! :) [2/25/2006 3:43 PM] 0 ;x [2/25/2006 3:43 PM] fuck [2/25/2006 3:43 PM] tehee [2/25/2006 3:43 PM] It always works if you press 1/x on my calculator. :( [2/25/2006 3:43 PM] Ch4r pointed out the flaw in your reasoning ;) [2/25/2006 3:44 PM] aw nuts [2/25/2006 3:44 PM] For x = 0, 1/x isn't defined :) [2/25/2006 3:44 PM] So, we can't say that the sentence is true. [2/25/2006 3:44 PM] *ahem* [2/25/2006 3:44 PM] d'you mean false? [2/25/2006 3:44 PM] *can't* [2/25/2006 3:45 PM] That's right that it is true for most of the numbers, but since it isn't true for the number 0 (being the exception), our sentence is mathematically false. :) [2/25/2006 3:45 PM] Elmo_, whoops -_- [2/25/2006 3:45 PM] And now, Narada, another one for you, tell me if this sentence is true : [2/25/2006 3:46 PM] oh goodie [2/25/2006 3:46 PM] "Every natural number is either odd or even" [2/25/2006 3:46 PM] *cough* [2/25/2006 3:46 PM] No, 0 is a natural number isn't it? err... I thought I remember it being called even before, but it is controversial, so I'm not sure. :/ [2/25/2006 3:47 PM] Natural numbers are 0 and up correct? [2/25/2006 3:47 PM] Yes ;) [2/25/2006 3:47 PM] So? True or false? [2/25/2006 3:47 PM] Well if 0 is even, it's true; if 0 is neutral, it's false. :p [2/25/2006 3:48 PM] 0 is even :) [2/25/2006 3:48 PM] wtf? [2/25/2006 3:48 PM] harr [2/25/2006 3:48 PM] Good 'ol fucked up memory saved me. [2/25/2006 3:48 PM] Yes Elmo, 0 is even :) [2/25/2006 3:48 PM] how so? [2/25/2006 3:48 PM] 1 is odd [2/25/2006 3:48 PM] That's just following a patern... [2/25/2006 3:49 PM] ... which is what defines if a number is even or odd. [2/25/2006 3:49 PM] I think. [2/25/2006 3:49 PM] yes, but, I'm going to stay a bit on these terms ;) [2/25/2006 3:49 PM] and explain. [2/25/2006 3:49 PM] We say that a number is 'even' if this number is divisible by 2. [2/25/2006 3:50 PM] (or, it is equivalent to say : a number is even if it's a multiple of 2) [2/25/2006 3:50 PM] We say that a number is 'odd' if this number gives a remainder of 1 when divided by 2. [2/25/2006 3:51 PM] (an equivalent statement is : an 'odd' number is a number that is follows an 'even' number) [2/25/2006 3:51 PM] So, if we call 'n' our natural integer, then we can say that : [2/25/2006 3:52 PM] # n is even <==> n = 2*k [2/25/2006 3:52 PM] # n is odd <==> n = 2*k + 1 [2/25/2006 3:52 PM] e.g. : 6 is even : 6 = 2*3 [2/25/2006 3:52 PM] 7 is odd : 7 = 2*3 + 1 [2/25/2006 3:53 PM] These two formulas are the key to understand properties of odd an even numbers! :) [2/25/2006 3:53 PM] -->| death_in_a_can (death_in_a@bu-5AC87799.sfldmidn.dynamic.covad.net) has joined #lecture [2/25/2006 3:53 PM] So, Elmo, you can see that : 0 = 2*0, hence, 0 is even ;) [2/25/2006 3:54 PM] got it? [2/25/2006 3:54 PM] yep [2/25/2006 3:55 PM] Alright, now, let's see basic operations with respect to odd and even numbers [2/25/2006 3:55 PM] 1° Addition (or substraction) ! [2/25/2006 3:56 PM] What happens if I add an even number with an other even number? [2/25/2006 3:56 PM] Let's see : [2/25/2006 3:56 PM] it'll be an odd [2/25/2006 3:56 PM] 2*k + 2*l = 2*(k+l) = 2*m (with m = k+l) [2/25/2006 3:56 PM] or not [2/25/2006 3:56 PM] So, it'll be an even number. [2/25/2006 3:57 PM] And it will ALWAYS be an even number ;) [2/25/2006 3:57 PM] Thus, Narada, is the following sentence is true? : [2/25/2006 3:57 PM] not again :/ [2/25/2006 3:57 PM] lmao [2/25/2006 3:57 PM] "An even number added to another even number will give an even number." [2/25/2006 3:57 PM] TRUE! [2/25/2006 3:57 PM] oops [2/25/2006 3:58 PM] Natural numbers? [2/25/2006 3:58 PM] Yes :) [2/25/2006 3:58 PM] True. [2/25/2006 3:58 PM] w00t! [2/25/2006 3:58 PM] but in fact, it also works if we take relative numbers ;) [2/25/2006 3:59 PM] e.g. : 4 + (-2) = 2 --> even + even = even [2/25/2006 3:59 PM] We just started doing nonreal numbers with calc in my math class :) [2/25/2006 3:59 PM] That funky i symbol. [2/25/2006 3:59 PM] ah yeah :) [2/25/2006 3:59 PM] complex numbers ;) [2/25/2006 3:59 PM] but, that's not the point of this lecture :p [2/25/2006 4:00 PM] Narada, what grade are you in? [2/25/2006 4:00 PM] 12 [2/25/2006 4:00 PM] :O [2/25/2006 4:00 PM] keep the random conversation in #binaryuniverse... on with the lecture ;) [2/25/2006 4:01 PM] Ok, so, who can apply my reasoning to the case 'odd + odd'? :) [2/25/2006 4:02 PM] = even [2/25/2006 4:02 PM] and even + odd = odd... ;/ [2/25/2006 4:03 PM] can you prove them? ;) [2/25/2006 4:03 PM] uh [2/25/2006 4:03 PM] good question;\ [2/25/2006 4:04 PM] teehee [2/25/2006 4:04 PM] Elmo, can you? :) [2/25/2006 4:04 PM] 3 + 3 = 6 || 9 + 9 = 18 = proven [2/25/2006 4:04 PM] no no! [2/25/2006 4:04 PM] oh.. you mean mathematically? [2/25/2006 4:04 PM] this is not a proof! [2/25/2006 4:04 PM] yes I do ;) [2/25/2006 4:05 PM] 2k-1 + 2l-1 = 2*(k+l)-1 [2/25/2006 4:05 PM] or something [2/25/2006 4:06 PM] lol, I told you an odd number was : 2*k+1, so use that :) [2/25/2006 4:06 PM] o [2/25/2006 4:06 PM] same poop [2/25/2006 4:06 PM] I know it's true that an odd number is : 2*k - 1 [2/25/2006 4:06 PM] but, it's better to have "+" than "-" ;) [2/25/2006 4:06 PM] 2k+1 + 2k+1 = 2(k+l)+1 [2/25/2006 4:07 PM] no! [2/25/2006 4:07 PM] hmm... [2/25/2006 4:07 PM] since this would give you : odd + odd = odd, which is not true ;) [2/25/2006 4:07 PM] 2k+1 + 2l+1 = 2(k+l)=1? [2/25/2006 4:07 PM] i give up... Elmo_ needs cookies [2/25/2006 4:07 PM] +1* [2/25/2006 4:07 PM] not = [2/25/2006 4:08 PM] that would be the same as Elmo's [2/25/2006 4:08 PM] oh ;/ [2/25/2006 4:08 PM] Let me do this one ;) [2/25/2006 4:08 PM] 2k+1 + 2k+1 = 2(k+l)+2 [2/25/2006 4:08 PM] because 1+1 = 2 [2/25/2006 4:08 PM] :-) [2/25/2006 4:08 PM] 2k + 1 + 2l + 1 = 2(k+l+1) = 2m [2/25/2006 4:08 PM] yes, Elmo got the 'trick' ;) [2/25/2006 4:08 PM] So : Odd + odd = even [2/25/2006 4:08 PM] Elmo_ got the style [2/25/2006 4:09 PM] and, Narada knows that this sentence is true ;) : [2/25/2006 4:09 PM] har har [2/25/2006 4:09 PM] "An odd number added with an odd number gives an even number" [2/25/2006 4:09 PM] yup [2/25/2006 4:10 PM] Ok, so now, Elmo, do the 'odd+even' case ;) [2/25/2006 4:10 PM] yes sir [2/25/2006 4:10 PM] 2k+1 + 2l = 2(k+l+1) = 2m+1 [2/25/2006 4:11 PM] i hope i'm right [2/25/2006 4:11 PM] your second step is false [2/25/2006 4:11 PM] yarr [2/25/2006 4:11 PM] oh [2/25/2006 4:11 PM] 2(k+l)+1 [2/25/2006 4:11 PM] because we need 1 not 2 :-) [2/25/2006 4:12 PM] ;) [2/25/2006 4:12 PM] so : odd + even = ? [2/25/2006 4:12 PM] odd [2/25/2006 4:12 PM] yep [2/25/2006 4:12 PM] and, since the '+' operator is commutative, we have that : [2/25/2006 4:12 PM] even + odd = odd [2/25/2006 4:12 PM] :) [2/25/2006 4:12 PM] short way [2/25/2006 4:13 PM] Yep, so, to sum it up, we have that : the addition of two numbers that are either both odd or both even gives an even number. [2/25/2006 4:13 PM] Does it remind you of some logic operator by any chance? :) [2/25/2006 4:13 PM] just like.. NN = P || PP=P [2/25/2006 4:14 PM] while NP = N [2/25/2006 4:14 PM] ? [2/25/2006 4:14 PM] Negative | Positive [2/25/2006 4:14 PM] a LOGIC operator [2/25/2006 4:14 PM] Xor :) [2/25/2006 4:15 PM] Yay! :D [2/25/2006 4:15 PM] So, now, let's study the "multiplication" [2/25/2006 4:15 PM] What happens if we multiply two even numbers? [2/25/2006 4:15 PM] even [2/25/2006 4:16 PM] Well, let's see! :) [2/25/2006 4:16 PM] 2 odds : odd [2/25/2006 4:16 PM] 2*k * 2*l = 2(k*l) --> even [2/25/2006 4:16 PM] ahem [2/25/2006 4:16 PM] sorry [2/25/2006 4:16 PM] it should be : [2/25/2006 4:17 PM] 2*k*2*l = 4*k*l = 2*(2*k*l) --> even [2/25/2006 4:17 PM] Now, prove the 'odd * odd' case Elmo ;) [2/25/2006 4:18 PM] okay... [2/25/2006 4:18 PM] 2k * 2l + 1 = 4kl + 1 = 2(2kl)+1 --> odd [2/25/2006 4:19 PM] no no [2/25/2006 4:19 PM] it's "odd * odd", not "even * odd" :) [2/25/2006 4:19 PM] So : [2/25/2006 4:19 PM] 2(k+1)*2(l+1) = 4(kl+2) [2/25/2006 4:19 PM] right? ;\ [2/25/2006 4:19 PM] ahem, no [2/25/2006 4:19 PM] ;o [2/25/2006 4:20 PM] (2k+1)*(2l+1) ;) [2/25/2006 4:20 PM] and, now... FOIL! :) [2/25/2006 4:20 PM] w00t! [2/25/2006 4:20 PM] >_< [2/25/2006 4:20 PM] so.. [2/25/2006 4:21 PM] 4kl + 2k + 2l + 1... [2/25/2006 4:21 PM] hmm [2/25/2006 4:21 PM] 4 + 2l + 2 +... [2/25/2006 4:21 PM] Ch4r's got the right one ;) [2/25/2006 4:21 PM] 4lk+2k+2l_1 [2/25/2006 4:22 PM] gah too late [2/25/2006 4:22 PM] Narada: I win. [2/25/2006 4:22 PM] -->| diac_bot (death_in_a@bu-5AC87799.sfldmidn.dynamic.covad.net) has joined #lecture [2/25/2006 4:22 PM] ;) [2/25/2006 4:22 PM] :p [2/25/2006 4:22 PM] so : 4kl + 2k + 2l + 1 = 2(2kl + k +l) + 1 --> odd [2/25/2006 4:22 PM] Hence : odd*odd = odd :) [2/25/2006 4:22 PM] And now, who's volunteer for : odd*even? :) [2/25/2006 4:23 PM] * Elmo_ points a finger to the right. [2/25/2006 4:23 PM] * Ch4r points a finger to the left [2/25/2006 4:23 PM] * Narada ducks [2/25/2006 4:23 PM] haha [2/25/2006 4:23 PM] Narada: too late :D [2/25/2006 4:23 PM] it's you qwertydawom [2/25/2006 4:23 PM] okay ;P [2/25/2006 4:24 PM] so : (2k+1)*(2l) = 4kl + 2l = 2(2kl + l) --> even :) [2/25/2006 4:25 PM] and now, who tells me why the case 'even*odd' will give the same result? [2/25/2006 4:25 PM] * Elmo_ is fucking [2/25/2006 4:25 PM] ducking* [2/25/2006 4:25 PM] damn [2/25/2006 4:26 PM] so? [2/25/2006 4:27 PM] i dunno [2/25/2006 4:27 PM] because be make the 1 into a 2 by foil [2/25/2006 4:27 PM] no no [2/25/2006 4:27 PM] because... the '*' operator is commutative ;) [2/25/2006 4:29 PM] Ok, so, to sum it up, we have that : [2/25/2006 4:29 PM] Only the multiplication of two odd numbers gives an odd product. [2/25/2006 4:29 PM] Does it remind you of a logic operator? :) [2/25/2006 4:29 PM] nor ? [2/25/2006 4:29 PM] nope [2/25/2006 4:29 PM] and [2/25/2006 4:29 PM] ah [2/25/2006 4:30 PM] yes ch4r ;) [2/25/2006 4:30 PM] and now, we shall see the particular case of the 'square' of a number. [2/25/2006 4:30 PM] I guess you all know what it is? [2/25/2006 4:31 PM] yeah [2/25/2006 4:31 PM] root? [2/25/2006 4:31 PM] no, just the square :) [2/25/2006 4:31 PM] not the square root [2/25/2006 4:31 PM] e.g. : 2 squared -> 4 [2/25/2006 4:31 PM] ah [2/25/2006 4:32 PM] x * x [2/25/2006 4:32 PM] yes [2/25/2006 4:32 PM] cube would be x^x right? (sorry, just a curious equestion) [2/25/2006 4:32 PM] so, what can we say about : even^2 [2/25/2006 4:32 PM] no, cube would be : x*x*x :) [2/25/2006 4:32 PM] ah [2/25/2006 4:33 PM] even^2 = even [2/25/2006 4:33 PM] for even * even = even [2/25/2006 4:33 PM] if my mind didn't mess up [2/25/2006 4:33 PM] don't you dare ask us to prove that qwerty :P [2/25/2006 4:33 PM] haha :) [2/25/2006 4:34 PM] and, what about : odd^2? [2/25/2006 4:34 PM] odd [2/25/2006 4:34 PM] yup :) [2/25/2006 4:35 PM] So, we can say that a number squared keeps its odd/evenness :) [2/25/2006 4:35 PM] Now, thanks to Elmo's curiosity, we shall study the case where we take the cube of the number. [2/25/2006 4:36 PM] it's the same as squaring a number, yes? [2/25/2006 4:36 PM] I mean as far as oddness/evenness [2/25/2006 4:36 PM] no? [2/25/2006 4:36 PM] oh, yes [2/25/2006 4:36 PM] agreed [2/25/2006 4:36 PM] and so is uhh... ^4 [2/25/2006 4:37 PM] and ^5 [2/25/2006 4:37 PM] ;x [2/25/2006 4:37 PM] so, Ch4r, what can you tell me about the oddness/evenness of a number raised to the n-th power? :) [2/25/2006 4:37 PM] the oddness/evenness of n is the same as n to any power :p [2/25/2006 4:38 PM] w00t! :) [2/25/2006 4:38 PM] if the power is greater than 0 [2/25/2006 4:38 PM] right/ [2/25/2006 4:38 PM] ?* [2/25/2006 4:38 PM] cuz 8^0 != even [2/25/2006 4:38 PM] yep :) [2/25/2006 4:38 PM] since : n^0 = 1 -> odd ;) [2/25/2006 4:39 PM] Alright now, tell me which other operation we're missing? [2/25/2006 4:40 PM] hmm [2/25/2006 4:40 PM] So far we've covered : addition, substraction, multiplication [2/25/2006 4:40 PM] and division, haven't we? [2/25/2006 4:40 PM] and exponents.. [2/25/2006 4:41 PM] no, we haven't discussed division :) [2/25/2006 4:41 PM] oh >_< [2/25/2006 4:43 PM] So, let's do it now ;) [2/25/2006 4:43 PM] What happens if we divide two even numbers? [2/25/2006 4:43 PM] 2k/2l = k/l, can we conclude? [2/25/2006 4:44 PM] odd [2/25/2006 4:44 PM] or even [2/25/2006 4:44 PM] either... O.o [2/25/2006 4:44 PM] yeah... [2/25/2006 4:44 PM] because, for instance, 24/12 = 2, which is even, but 24/24 = 1, which is odd :P [2/25/2006 4:45 PM] well [2/25/2006 4:45 PM] excluding the number its self... [2/25/2006 4:45 PM] 24 / 12 = 2 24 / 6 = 4 24 / 4 = 6... [2/25/2006 4:45 PM] so.. it's always even UNLESS its the number devided by its self [2/25/2006 4:45 PM] right qwertydawom ? [2/25/2006 4:45 PM] hmm [2/25/2006 4:45 PM] no :) [2/25/2006 4:45 PM] damn [2/25/2006 4:45 PM] fractions.. [2/25/2006 4:46 PM] ;x [2/25/2006 4:46 PM] we CANNOT conclude ;) [2/25/2006 4:46 PM] [INFO] Now logging to . [2/25/2006 4:46 PM] [INFO] Channel view for ``#lecture'' opened. [2/25/2006 4:46 PM] -->| YOU (D31337) have joined #lecture [2/25/2006 4:46 PM] =-= Topic for #lecture is ``Next logic lecture: Saturday, February 25th at 9 PM GMT +0 | /join #binaryuniverse'' [2/25/2006 4:46 PM] =-= Topic for #lecture was set by Ch4r on Sunday, February 19, 2006 7:16:08 PM [2/25/2006 4:46 PM] ok [2/25/2006 4:46 PM] good [2/25/2006 4:47 PM] yep [2/25/2006 4:47 PM] I think I already am. [2/25/2006 4:47 PM] for odd/odd, we can't conclude [2/25/2006 4:47 PM] Narada, ok, cool [2/25/2006 4:47 PM] shit, i'm not [2/25/2006 4:47 PM] i'll just copy and paste [2/25/2006 4:47 PM] it's ok [2/25/2006 4:47 PM] Delly logged [2/25/2006 4:47 PM] -->| nslain (nslain@bu-144BB6A8.hsd1.mn.comcast.net) has joined #lecture [2/25/2006 4:47 PM] k [2/25/2006 4:47 PM] yes Ch4r, we can't conclude :) [2/25/2006 4:47 PM] and, what about : odd/even, even/odd? [2/25/2006 4:48 PM] we can't conclude ;o [2/25/2006 4:48 PM] devision sucks... it's so unconstant [2/25/2006 4:48 PM] aye [2/25/2006 4:48 PM] Now, Elmo, you talked about the 'square root' [2/25/2006 4:48 PM] What about it? [2/25/2006 4:48 PM] sqrt(even)? [2/25/2006 4:48 PM] sqrt(odd)? [2/25/2006 4:49 PM] Are evens even and odds odd? [2/25/2006 4:50 PM] sqrt(2)? :p [2/25/2006 4:50 PM] gah [2/25/2006 4:50 PM] sqrt(even) = odd [2/25/2006 4:50 PM] nah [2/25/2006 4:50 PM] sqrt(81) = 9 [2/25/2006 4:50 PM] even = even [2/25/2006 4:50 PM] teehee [2/25/2006 4:51 PM] no [2/25/2006 4:51 PM] since x^2 has the same evenness/oddness as x, the square root of x^2 has to have the same evennness/oddness as x^2... [2/25/2006 4:52 PM] cannot conclude? [2/25/2006 4:52 PM] elmo's right ;) [2/25/2006 4:52 PM] Ch4r: that's what i thought [2/25/2006 4:52 PM] oh wait... [2/25/2006 4:52 PM] yeah, I see [2/25/2006 4:52 PM] Ch4r, you know what's wrong? :) [2/25/2006 4:52 PM] yeah, I see. [2/25/2006 4:52 PM] * Narada explodes. [2/25/2006 4:52 PM] It's a simple implication ;) [2/25/2006 4:52 PM] x even/odd => x^2 even/odd :) [2/25/2006 4:55 PM] gotcha.. [2/25/2006 4:56 PM] in fact, your reasoning would work with integers ;) [2/25/2006 4:56 PM] if we know that n is the square of an integer [2/25/2006 4:57 PM] then, we can say that sqrt(n) has the same evenness/oddness than n :) [2/25/2006 4:57 PM] ok? [2/25/2006 4:58 PM] mhm [2/25/2006 4:59 PM] e.g. : "4 is the square of an integer", so, we know that sqrt(4) will be even :) [2/25/2006 5:02 PM] * Ch4r waits :p [2/25/2006 5:03 PM] Ok lol [2/25/2006 5:03 PM] So, we've seen that we couldn't conclude with the square root [2/25/2006 5:05 PM] And, this ends this (long) parenthesis about odd and even numbers ;) [2/25/2006 5:05 PM] Now, let's get back to the concept of 'truth' [2/25/2006 5:06 PM]