For those who are confused about why the Cartesian product of {0, 1} with {a, b} is equal to {(0, a), (0, b), (1, a), (1, b)} and not equal to {(0, a), (0, b), (1, a), (1, b), (a, 0), (a, 1), (b, 0), (b, 1)} is because, by definition, the Cartesian product demands that the elements (a, b) in the set Cartesian(A, B) be such that a is from set A and b is from set B. This definition was directly stated in the video too, so it is not as though Dr. did not explain it properly. He definitely did explain it.
Why can’t more teachers be like you? My professor just presents the information instead of teaching it or trying to help us understand it. You’re a good man. Thank you.
Cause in school you cant choose you professor whereas here on youtube u can choose between many so thats why He is Great at Teaching , because it competitive (for views)
Thanks a thousand times for this video series, I missed a couple of my uni lectures due to illness and these have been invaluable in helping me catch up. Very thorough succinct and well explained!
I can't thank u enough I was stressed the whole day because I couldn't understand it and then this video just saved my day from being unproductive 😂😂😇 I can't thank u enough sir love from India
I have had good time to follow this credible lectures by professor, a lot of thanks but i need to know more. May Almighty God blessed you more and more Sir.
What a professional teaching video,I really appreciate your efforts,you have no idea how much enlightenment you put on students path towards learning and understanding the subject by your teaching skills.
nice video! In the ordered pair, does it matter which order the elements are in? So that {a,b}X{0,1} would be {(a,0),(a,1),(b,0),(b,1),(0,a),(1,a),(0,b),(1,b)}
It does matter, but the product you calculated is incorrect. Namely, Cartesian({a, b}, {0, 1}) = {(a, 0), (a, 1), (b, 0), (b, 1)} is the correct calculation. What you calculated is actually equal to Union(Cartesian({a, b}, {0, 1}), Cartesian({0, 1}, {a, b})). It should be remembered that the Cartesian product is not symmetric, or in other words, not commutative, and it is not associative either. In fact, it is not even flexible or power associative.
I know this video is old, but I think there would be merit in revisiting the topic with a more elaborate series of videos, one which, in particular, expands more on what exactly is being meant by an "ordered pair," in terms of set theoretic definitions. For example, I have seen different authors use different definitions for the Cartesian product. Some authors define it as the set of all functions f with domain {0, 1} such that f : 0 |-> a, where a is an element of A, and f : 1 |-> b, where b is an element of B. The reason for this difference in definition and this type of complication may ultimately have to do with how you choose to define ordered pairs and tuples in mathematics. For example, some authors will define all ordered pairs as Kuratowski pairs, with (x, y) := {{a}, {a, b}}. However, this definition is not very useful for defining higher order n-tuples or generalizing without creating problems an inconsistencies. On the other hand, some authors define n-tuples as functions f : {m : 0 =< m =< n} -> S, with S being an arbitrary set, while functions are themselves defined as sets containing Kuratowski pairs {{x}, {x, y}}, such that x is an element of the domain, and y is an element of the codomain, with y being unique to every x. In this alternative convention, Kuratowski pairs are different from general 2-tuples, which can make it more complicated to define some objects more precisely, since now you have to be more specific, but this has the advantage that this generalizes n-tuples uniformly and consistently, and it even works for the 0-typle and 1-tuples, and it also makes the definition of set exponentiation work better. However, as this is quite a complicated nuance that can confuse people with regards to what types of ordered pairs are being considered in different Cartesian products and what consequences this has, this definitely merits a video series on.
This is a great overview of the Cartesian product, thank you Trefor Bazett. As a suggestion, it might be great to have some kind of “accompany overview” that outlines how much depth of a certain topic was actually covered and what are keywords, thoughts, proofs that have been left out. This could be just a link to good material, I am currently mostly trying to look a topic up on Wikipedia to understand the overall scope better, but I am certain there are more suitable overviews out there. Thank you for your excellent work!
If the order is important in pairs so (a,b) != (b,a).. so why in the set of the cartesian product in the example you write (a,0) but you didnt write (0,a)?
letA=(6,7,8,10,12,13)and B=-(5,7,9,11,13,15) and the relation R from A to B means,"is greater than".find R.Also ,draw a suitable diagram to represent this relation .. sir plese help me with this question
In the beginning, you said that order matters so at the last example should contain eight order pairs instead of four. and by the way you're way of teaching is so great! and this course is super beneficial.
No, you are incorrect. What he wrote in the end is correct. The Cartesian product of A with B is not the same as the Cartesian product of B with A, and the answer with eight elements you are suggesting is a union of both products, not the actual Cartesian product of A with B.
The definition of the Cartesian product as stated in the video, which is also the definition that every mathematician uses, is that Cartesian(A, B) contains all the ordered pairs (a, b) such that, specifically, a is an element of A, and b is an element of B. So if b is an element of B, but not an element of A, then (b, a) is NOT an element of Cartesian(A, B). This is why you are wrong.
I have a question if I may. I am struggling to find the root reason as to why we need such a concept in the first place (The concept of ordered-pair). Is it because we needed it for 2-dimentional spatial plotting in the cartesian coordinate system? Or was there another necessity that made us come up with such concept? Thank you.
Functions are defined as sets of ordered pairs, as are all binary relations and all graphs. Ordered pairs are an incredibly useful and even a practical idea applicable to, really, any field of study, even outside mathematics.
for the cartesian product problem, why doesn’t the product include different orders of the pairs, since order matters in this case. ex. why isn’t (1,a) (2,a).... included
(1, a) and (0, b) are not elements of Cartesian(A, B), but of Cartesian(B, A). Remember that, by definition, the first element of the ordered pairs (a, b) that are elements of Cartesian(A, B) are always elements of A.
@@mangomilker9227 It's not surprising. Let's say that there is a 0.01% chance that someone who watches this video is a 7th grader.. There are 21k views on this account. The odds of zero 7th graders watching this video is (0.9999)^(21000), or 12%. That means there's an 88% chance that at least one 7th grader watches this video.
@@gamerdio2503 That calculation makes absolutely no sense. For one, most viewers of the channel are most likely not subscribers, as is true of most small TH-cam channels. For two, if the probability of someone watching the video being a 7th grader is 0.0001, then the probability of a 7th grader being among the 21K subscribers is (0.0001)^(21000), not (0.9999)^(21000).
@@angelmendez-rivera351 1. But we are not concerned with subscribers, only viewers. Therefore the subscribers dont matter. 2. (0.9999)^(21000) is the odds of NO 7th grader watching the video. 1 - that probability tells us the odds of AT LEAST 1 7th grader.
For those who are confused about why the Cartesian product of {0, 1} with {a, b} is equal to {(0, a), (0, b), (1, a), (1, b)} and not equal to {(0, a), (0, b), (1, a), (1, b), (a, 0), (a, 1), (b, 0), (b, 1)} is because, by definition, the Cartesian product demands that the elements (a, b) in the set Cartesian(A, B) be such that a is from set A and b is from set B. This definition was directly stated in the video too, so it is not as though Dr. did not explain it properly. He definitely did explain it.
i feel like this overall video isn't the best.
but I did eventually come back to this video after watching some other ones. now it seems more obvious.
Lg. 😮rg
@@raymondmann9947 What? 🤔
Why can’t more teachers be like you? My professor just presents the information instead of teaching it or trying to help us understand it.
You’re a good man. Thank you.
same
Same
Cause in school you cant choose you professor whereas here on youtube u can choose between many so thats why He is Great at Teaching , because it competitive (for views)
Thanks a thousand times for this video series, I missed a couple of my uni lectures due to illness and these have been invaluable in helping me catch up. Very thorough succinct and well explained!
THANK YOU SO MUCH FOR MAKING THIS! My professor is old school and is teaching from the textbook. The way you teach makes it easy to understand!!!
I can't thank u enough I was stressed the whole day because I couldn't understand it and then this video just saved my day from being unproductive 😂😂😇 I can't thank u enough sir love from India
Glad it helped!
"a & b could come from different sets."
Mind blown...
Used the Cartesian coordinate system for a long time, I never really understood what it was.
u r awesome sir .... ur way of teaching is awesome
I went through a few videos to see if you fixed the echoing sound. Thank goodness you did it quickly. I will be watching this course. Thanks!
and the oscar for most dramatic computer science teacher goes tooooooo
You make math fun! Makes me we want to learn more!
Sir what is the cartesian product a*b if a ={}and b={a,b,c}
The empty set
@@DrTrefor thanks sir 🙏🙏 thanku so much 🙏🙏
I have had good time to follow this credible lectures by professor, a lot of thanks but i need to know more. May Almighty God blessed you more and more Sir.
Just wow. This is such a clear explanation!
Impressively high-quality video lesson👍👍👍
Thank you so much from a very very very very grateful undergrad.
So glad it is helping, thanks for becoming a member! :)
Thanks. This helps me reading sean caroll's book on general relativity.
What a professional teaching video,I really appreciate your efforts,you have no idea how much enlightenment you put on students path towards learning and understanding the subject by your teaching skills.
Im so grateful I found this channel, props to you Dr!
Thank you!
I have been listening to these in preparation for my final. Here’s to hoping I get this! Thanks for the good teaching ^-^
At 6:52 I think he uses the term "cross product" twice when he means Cartesian product.
nice video! In the ordered pair, does it matter which order the elements are in?
So that {a,b}X{0,1} would be {(a,0),(a,1),(b,0),(b,1),(0,a),(1,a),(0,b),(1,b)}
It does matter, but the product you calculated is incorrect. Namely, Cartesian({a, b}, {0, 1}) = {(a, 0), (a, 1), (b, 0), (b, 1)} is the correct calculation. What you calculated is actually equal to Union(Cartesian({a, b}, {0, 1}), Cartesian({0, 1}, {a, b})). It should be remembered that the Cartesian product is not symmetric, or in other words, not commutative, and it is not associative either. In fact, it is not even flexible or power associative.
Awesome video. "and do lots of practice problems" where could one get some from?
You are a legend. I am deeply grateful to you.
exlained the concept by applying it to he real world very well sir
Honestly a great and succinct video and explanation - Thanks!
Thank you Dr Bazett for this amazing gift! What text would you recommend for problem solving and examples?
Wish my engineering maths teachers taught like you. Thanks for all the videos. Discovering the joy of learning maths decades after finishing college.
sir thank you for your these vedioes.It was help for my studies
So can you just basically use distributive property across the elements of the sets?
I have a Formal Systems & Logic midterm in 9 hours, you are like an angel who has descend from heaven to gift me a passing grade
Thanks. If you want to go quickly, putting it at 1.25x speed to maybe 1.5(provided you concentrate hard and know a bit of the syllabus) helps.
I know this video is old, but I think there would be merit in revisiting the topic with a more elaborate series of videos, one which, in particular, expands more on what exactly is being meant by an "ordered pair," in terms of set theoretic definitions. For example, I have seen different authors use different definitions for the Cartesian product. Some authors define it as the set of all functions f with domain {0, 1} such that f : 0 |-> a, where a is an element of A, and f : 1 |-> b, where b is an element of B. The reason for this difference in definition and this type of complication may ultimately have to do with how you choose to define ordered pairs and tuples in mathematics. For example, some authors will define all ordered pairs as Kuratowski pairs, with (x, y) := {{a}, {a, b}}. However, this definition is not very useful for defining higher order n-tuples or generalizing without creating problems an inconsistencies. On the other hand, some authors define n-tuples as functions f : {m : 0 =< m =< n} -> S, with S being an arbitrary set, while functions are themselves defined as sets containing Kuratowski pairs {{x}, {x, y}}, such that x is an element of the domain, and y is an element of the codomain, with y being unique to every x. In this alternative convention, Kuratowski pairs are different from general 2-tuples, which can make it more complicated to define some objects more precisely, since now you have to be more specific, but this has the advantage that this generalizes n-tuples uniformly and consistently, and it even works for the 0-typle and 1-tuples, and it also makes the definition of set exponentiation work better. However, as this is quite a complicated nuance that can confuse people with regards to what types of ordered pairs are being considered in different Cartesian products and what consequences this has, this definitely merits a video series on.
Thank you kind sir for your videos. You've gained a subscriber.
thank you,,,,💘
This is a great overview of the Cartesian product, thank you Trefor Bazett. As a suggestion, it might be great to have some kind of “accompany overview” that outlines how much depth of a certain topic was actually covered and what are keywords, thoughts, proofs that have been left out. This could be just a link to good material, I am currently mostly trying to look a topic up on Wikipedia to understand the overall scope better, but I am certain there are more suitable overviews out there. Thank you for your excellent work!
he put a link to a nice online book in the first video of the series
Question: if i have (A × B)^2 how would this look like for example:
A = {1,2}
B = {3,7}
({1,2}×{3,7})^2
Good to hear from you and thanks for letting
I didn't skip the Ads because I want them to get budget and make more videos
Deserving way more likes
What separates the cartesian product from the power set?
Or are they the same thing?
In the Cartesian product we are only taking pairs of things, one form the first set one from the second
If the order is important in pairs so (a,b) != (b,a).. so why in the set of the cartesian product in the example you write (a,0) but you didnt write (0,a)?
Bro your beard looks great, the video was helpful as well. Many thanks :)
you open my eyes sir , awesome teaching sir!!
Sir , if possible ,please make videos on complex analysis
letA=(6,7,8,10,12,13)and B=-(5,7,9,11,13,15) and the relation R from A to B means,"is greater than".find R.Also ,draw a suitable diagram to represent this relation .. sir plese help me with this question
In the beginning, you said that order matters
so at the last example should contain eight order pairs instead of four.
and by the way you're way of teaching is so great!
and this course is super beneficial.
@@DrTrefor thanks a lot you're awesome!
No, you are incorrect. What he wrote in the end is correct. The Cartesian product of A with B is not the same as the Cartesian product of B with A, and the answer with eight elements you are suggesting is a union of both products, not the actual Cartesian product of A with B.
The definition of the Cartesian product as stated in the video, which is also the definition that every mathematician uses, is that Cartesian(A, B) contains all the ordered pairs (a, b) such that, specifically, a is an element of A, and b is an element of B. So if b is an element of B, but not an element of A, then (b, a) is NOT an element of Cartesian(A, B). This is why you are wrong.
Wouldn't you also include (0, a), (0, b), (1, a), and (a, 1) since these are *ordered* pairs?
If (a, 1) is not same as (1, a), should it not be one of the ordered pairs in the cartesian product?
Considering base rates and style of recording the video, what is the probability of Trefor being left-handed? 🙃
Does this cross have anything to do with the cross we learned in linear algebra?
Nope, completely different
Thanks very much sir helped me a lot
I have a question if I may. I am struggling to find the root reason as to why we need such a concept in the first place (The concept of ordered-pair). Is it because we needed it for 2-dimentional spatial plotting in the cartesian coordinate system? Or was there another necessity that made us come up with such concept? Thank you.
Functions are defined as sets of ordered pairs, as are all binary relations and all graphs. Ordered pairs are an incredibly useful and even a practical idea applicable to, really, any field of study, even outside mathematics.
I'm already finishing up my first month of computer science courses.
Great teacher!
for the cartesian product problem, why doesn’t the product include different orders of the pairs, since order matters in this case. ex. why isn’t (1,a) (2,a).... included
Because the Cartesian product is not commutative. The Cartesian product of A with B is not the same as the Cartesian product of B with A.
so, im confused. cartesian pairs are elements of a set and not sets themselves?
do you have a proof on (a,b)=(c,d) iff a=c and b=d
I would say this is a definition, not a proof. Otherwise, what do you mean by (a,b)=(c,d)?
this man writing backwards for 83 videos for the benefit of mankind
lol, I wondered about that while watching his video for the first time. Wondered if he was writing backwards or just flipped the video image.
If order matters in ordered pairs, why are there only 4 pairs in the A x B example? Why is {1, a} or {0, b} not part of the set?
@@DrTrefor what do you mean by component?
@@tennenyt5311 { (a, b)}
The a is first component b is the second one. (a, b) is the element. { (a, b) } is the set.
Correct me if I'm wrong
@@kems1592 You are correct
(1, a) and (0, b) are not elements of Cartesian(A, B), but of Cartesian(B, A). Remember that, by definition, the first element of the ordered pairs (a, b) that are elements of Cartesian(A, B) are always elements of A.
you are very smart sir
He finally gets to the point at 5:13
So, can we say that in Python list(zip([1,2,3], [5,6,7])) produces a cartesion product?
No. Take the case of {a, b} x {0, 1}. list(zip([a, b], [0, 1])) returns [(a, 0), (b, 1)]. It does not contain (a, 1) or (b, 0)
I like your shirt :)
I want to ask if (a,4)and (4,a) are same
They are not the same unless a == 4.
What is the first element of (2,5)
2
Plzzz answer it because I am going to do my assignment
😊
:46 "I want to have the first one of them in specificed in the second one of them is also going to be specified" huh?????
he said and not in
Poli sanam
tuples!
Happy Star Wars Day. (upload date)
:46 to :51 I didn't understand what you said
Hi:
How can I come back to the video I left off at?
Hm interesting
So it's not commutative
Yeah I see all the other comments but I’m literally a 7th grader lol I at least get 70% of this
I wonder how surprising it is to see a 7th grader her I just want to make computers (when I’m older)
Trefor Bazett lol
@@mangomilker9227 It's not surprising. Let's say that there is a 0.01% chance that someone who watches this video is a 7th grader.. There are 21k views on this account. The odds of zero 7th graders watching this video is (0.9999)^(21000), or 12%. That means there's an 88% chance that at least one 7th grader watches this video.
@@gamerdio2503 That calculation makes absolutely no sense. For one, most viewers of the channel are most likely not subscribers, as is true of most small TH-cam channels. For two, if the probability of someone watching the video being a 7th grader is 0.0001, then the probability of a 7th grader being among the 21K subscribers is (0.0001)^(21000), not (0.9999)^(21000).
@@angelmendez-rivera351 1. But we are not concerned with subscribers, only viewers. Therefore the subscribers dont matter.
2. (0.9999)^(21000) is the odds of NO 7th grader watching the video. 1 - that probability tells us the odds of AT LEAST 1 7th grader.
а як ти дзеркально пишеш альо
load of waffle -get to the point
you are very smart sir