Check out my new course in Set Theory: trevtutor.com/p/master-discrete-mathematics-set-theory It comes with video lectures, text lectures, practice problems, solutions, and a practice final exam!
It's almost like those universities don't really *want* you to learn. It's infuriating how much potential is lost by that. I'm in my 30s now and I always thought I hated math, but this is actually a nice little puzzle and logic game instead of panic and confusion.
Thank you so much for your videos. I went to a school where we were divided into groups dependent on maths ability and once in your group you could never really learn above that level. My lack of maths skills i know is down to the teachers i had and that they didnt make it fun and understandable. Now as a 33 year old im feeling mature enough to try again and try and get a degree in computer science. Your videos make me feel like it might be possible for me to master maths and actually enjoy it.
I came here after realizing there was no playlist for Discrete Math for the Organic Chemisty Tutor. I’m happy to report I’m equally pleased with your videos so far. Thanks, dude!
Great lectures, thank you. I've been a software engineer for 10+ years, but hoping to get a more rigorous understanding of computer science so I'm starting with discrete maths.
Great! Thanks Trev. It would be great if the volume of your videos were just a tad bit louder. I had to max system and RU-vid volume just be to able to hear you, and I'm sporting a pair of 6'' studio monitors that regularly get noise complaints...
Why are Cartesian coordinates represented as {{x}, {x,y}}? Is it just so that you can differentiate between two ordered pairs that have the same x and y, but flipped?
Sets by definition have no predefined order according to the previous video. This is why the coordinates are represented specifically as ordered pairs. The first variable stated (in an ordered pair) dictates the order of the pair.
at 8:06, isn't (c,d) and (d,c) the same because set doesn't have order, also if then zoom out, the larger set has two same set of (c,d), (d,c), which would conclude BxB = {(c,c),(c,d),(d,d)}?
thank you sir for making this vlog it really helps me for being an freshman student in Bachelor of science and information course. please im looking forward for more
if A = { 0, 1} and B = { 2,3,4 } What is A x A x B? If I write out all the possible combinations it's: { {0,1}, {0,1}, {0,2}, {0,3}, {0,4}, {1,2}, {1,3}, {1,4} } = 8 But A x A = 4 and A x B = 6 which is 12. Am I doing something wrong?
I'm wondering what the link is between {a, {ab}} and a Cartesian product? None of the Cartesian products have subsets of different lengths so I don't see the connection
I have to agree with comments below. I had to take a course at the the university I attended for my BS degree. The text book was "Elementary Functions" that addressed Cartesian products. The class professor and text were a waste. They should have name the course deciphering bullshit. Thanks for making this easy.
Regarding Ø × A, I understood the cardinality argument, that it's size should be zero, so the result is Ø. Taking that detour makes sense. But what threw me off was, if you carried through the process as shown prior to this example, we would get { } × { a, b } = { ( , a), ( , b) }. Is this wrong or do we define the size of ( , a) to be zero? Or is (m, n) only defined if both elements are present? If something like ( , n) is not defined, how is the number zero derived from it? Thanks!!!
( ,a) is in fact not defined, i.e. does not correspond to any object. It is similar to how you can write x∈{}, but x does not correspond to any object. I think another way to see this is as follows. We can say that t∈AxB iff there are some a∈A and b∈B, such that t=(a,b). This means that t∈{} x {0,1} is equivalent to saying that there are some a∈{} and b∈{0,1} such that t=(a,b). However, no such a exists, and thus no such t exists either. This means that no element is in {} x {0,1}.
I'm not sure why do you write a singleton when we have a set for example (1,2) why did you write the singleton {1} then the set {1,2} I hope I'm making sense.
Hello i have a question on minute 8:32, would BxB be ( (c) , (c,d) , (d) ) since in the last video it was explained that repeated elements are listed once and theres no order in a set meaning that c,d is the same as d,c? just a bit confused 😅 Thanks
The set is unordered, but individual elements in cartesian products are ordered. If you see {, } it's unordered. If you see (, ) or then it's ordered and we must preserve that order.
I can't see the relation with the cartesian product as you define it and as it is defined in linear algebra, with vectors. Is there any? Actually, in linear algebra AxA would be zero.
Thank you Sir; but I have a question; I answered the question ∅×A as ( {a} , {a,b} ) but got it wrong . I did this because i thought that the null set doesn't count hence we treat set A alone as the ordered pair. Please what am I missing?
An empty set has a cardinality (size) of zero. When we Multiply two sets, the formula is the product of the cardinality of the 2 sets. A and B both has 2 a cardinality of two. 2 x 2 is 4, so you get 4 ordered pairs. In the case of an empty set, it becomes 0 x 2 which is 0, thus you get no ordered pair. Therefore, an empty set. Hope that answered your question.
For anyone reading this later on: those elements written in normal parentheses are ordered pairs, so their order still matters and thus they aren't the same. If you wanted to represent those ordered pairs as sets then you'd write {{c}, {d,c}} and they still wouldn't be the same thing.
How do i solve something like this? For each of the following pairs on the natural numbers N (in other words, elements from N x N), list the ordered pairs in each set. R = {:2x+y=9} S = {:x+y=7}
Well it appears to me that since natural numbers are positive integers between 1 and infinity, the answer to S should be {(1,6), (6,1), (2,5), (5,2), (4,3), (3,4)} because those are the pairs of natural numbers that add up to 7. For R I would look at it the same way starting with X values. If x is 1 then y would be 7 to equal nine, if x was 2 then y would be 5 to equal 9 so on and so forth. Therefore R = {(1,7), (2, 5), (3,3), (4,1)}. I may be completely off but that is how I logic through it.
What would be the result to the following A^13 * A^ 21 = ? Do we follow the exponent "product rule"? If so does this sound right to you A^13 * A^ 21 = A^34 !! please someone explain to me
Yes. Because as he showed in the video, |A| = m, |B| = n, and |A×B| = m ⋅ n. Since |A²| is just another way of writing |A×A|, |A²| = |A| ⋅ |A|, in other words, |A²| = |A|². For |A×A×A|, if we look at it as |(A×A)×A|, we get |A×A×A| = |A×A| ⋅ |A| = |A| ⋅ |A| ⋅ |A| = |A|³, so by the exponent properties it follows that |Aˣ| = |A|ˣ, so at that point we can just use them to show that |A¹³×A²¹| = |A¹³| ⋅ |A²¹| = |A|¹³ ⋅ |A|²¹ = |A|³⁴.
Thank you....I have a question. If S is the set of real number and I be set of rational number. Aalpha( the alpha is the symbol, but there is no symbol for alpha on my phone pad here. moreover the alpha is index set of the family. Aalpha = { x € S: x is greater than or equal to alpha} for any alpha is an element of index set. show that UAalpha = S.
Why are the empty set crosses equal to the empty set when you can obviously pair with the second set? The cross product would just result in the ordered pairs with only the second set's elements, which is obviously not an empty set.
Why you said cross I think it is wrong because of these definition : the Cartesian product is a set (of vectors), the cross product is a vector So you can’t say cross it is not the same you have to just say product. If I am wrong explain it to me please. And thank you so much for your videos 🤝
Yati Kasera I think it is because a nul set is an empty set/equal to zero. When you multiply anything with zero your answer will always be zero, or empty in this case
Cartesian products, 5 mins in, (0,0) looks like a face. Okay, w/e. Then all the other pairs start to look like faces. I think its time for a nap. Be back soon
On the first video dealing with an intro to sets, I was wondering whether the set "Desk" might not be best understood as belonging to the set "noun" since the desk is not a set that may contain itself as one of the elements. Just as "humans" is the set of specific mammals, but it itself is not a human, or, an element of itself. I am asking because in the example it is difficult to see, that might be the point, any relation between elements and sets. With numbers is slightly less difficult, but with linguistic examples, I thought that words belonging to logical-algebraic categories of language might be better.
Hey, I noticed that you mentioned linguistics. I am currently studying linguistics to better help me understand maths. I’m currently using crash course linguistics to study. Do you recommend any other materials that are still simply enough written but dives much deeper into linguistics?
@@gloriakalengelayi8294 From what I have seen Saussure opened up (also psychoanalysis actually) the field in terms of introducing the phonetic aspect. If one has taken classes in other languages in school, or, is bilingual one realizes how the phonetic aspect (out of it) one can begin to derive the words of other languages. The latter element is crucial to answer the question of he way human language is universal. The only material on ("generative"/Godel is the censored word used. It's actually dialectics) linguistics that I'm aware of that is worth taking a look at is that of Saussure, then all the three branches of formal logic and dialectical logic. It's difficult to find good material on the latter since it has been censored/mediated vía Kurt Godel's work on generativity. The only crucial works on "generative"/grammar (introductions) that I've read are in Spanish. The books were written by someone called Eduardo Vázquez. One of the crucial ones can be found on pdf on the net. The book is called "Los Puntos Fundamentales de la Filosofía de Hegel". He has others and can be found on the net. The stuff presented by this called called the "Einstein of Linguistics" doesn't look like anything. After being crowned with this title he was asked to at least (Einstein apparently had to) present his general theory. He was not able. After this individual retired he was asked again about the general theory that would acct for him being designated someone with a general theory of human language. Once again he claimed that he did not have it. That it was too difficult etc.
@@gloriakalengelayi8294 In a set of questions posed to this individual that goes by the name Chomsky he claimed that he had studied the phonetic (as well as other [differential]) element introduced by structural linguistics, but that he could not see the relevance to it regarding questions of "generativity" and the universality of language. The lecture and the set of questions can be found in RU-vid.
@@gloriakalengelayi8294 I would recommend presentations of Bertrand Russell's paradox in RU-vid. The looping logic is crucial to grasp the self-reflexivity of dialectics. Along with Vazquez's books I would recommend a book presented by Rebecca Goldstein and another by Todd McGowan. Goldstein's book is called "Incompleteness" and introduces the looping/generative/dialectical logic presented by Godel as his own. McGowans book is called "Emancipation After Hegel". The crucial thing is to concentrate upon formal and dialectical logic since that's what German Idealists were working with. McGowans propositions about contradiction begin with how they are found in language and then in the logic of people's practices etc.
