The only place where I found a clear breakdown of the definition of ordered pairs using sets. I couldn't wrap my head around the fact 'why is the definition like = {{x}, {x,y}} and not = {{x},{y}}, but this video showed me why. Thanks for this tutorial. Many regards.
Thanks to this video series I am beginning to understand why mathematicians like Frege, Russel and Hilbert thought that set theory is the foundational starting point from which all the other branches of mathematics can be built. Before viewing this video series I had only the high school version of set theory as Venn diagram.So I wondered how the notion of sets could be used to derive or construct mathematical objects like ordered pairs and coordinate plane. Now I have a intuitive sense of how the rigorous definitions of an ordered pair and Cartesian product will lead to talking about functions and relations. It's hard to follow philosophical issues in mathematics without doing some actual math yourself, and this video series definitely makes an intimidating subject quite easy to understand. Part 3 was a tougher going than the previous two, but it was definitely worth it. Thanks!
John Ok Glad you enjoyed the video. One thing I really tried to do in this video was give a clear explanation of the proof that = iff x=u & y=v. I'm not sure if I succeeded but it's important to understand in order to see that this description of ordered pairs using sets actually works in preserving the "ordering" property we want. By the way, ordered triples have a similar description, where (an ordered triple) is first represented as (an ordered pair) which can then, of course, be represented as a set, as can any other ordered pair. Then, you can abstract it to n-tuples (n is an integer) by nesting them in lower order tuples.
Very helpful video! This series has been changing set theory from vague concepts that I largely just memorize into a more logical and intuitive way of thinking. Awesome!
I've never seen the set theoretical justification for ordered pairs. It really makes a lot of sense and helps me to understand the construction of coordinate systems; so that's awesome. And that diagram at the end was freaking sweet. Combine this with metric spaces and you get something way cooler: linear algebra.
Thanx a lot for those lectures on set theory . I enjoyed every second , this is how mathematics should be explained . Keep the good work up . Many thanx .
wow, this was the best explanation of the kuratowski ordered pair definition Ive seen. I mean, most proofs with this def usually follow closely the same lines you did, but Ive never seen a derivation in which where you created that conjunction statement explicitly before.It was elegant because once youve eliminated the innermost two cases, then the two outermost automatically follow by AND. Btw, just found this series, and I'm looking forward to it. Been reading Enderton's set theory and Ian Stewart/David Tall's Foundations, and I have a real hard time grasping the proof of the Recursion Theorem.
Hello, sir. Do you presume that most of your practice problems have solutions online? It would be nice if you had a video in which you went through them all in sequence, although I know what a burden that would be.
+Mitchell Acampora It's a tradition in higher mathematics to propose practice problems and never give you the answers. But if you were curious about any of them, please ask.
Mathoma, it seems to me that the benefits of such "tradition" don't warrant the risk of perfecting a flawed approach to attacking a problem. At any rate, given the nature of the medium via which you present your lectures such information is much more conducive to instilling confidence in your audience and reaffirming their grasp on the subject matter, no?
+Samuel Cramer You're welcome and I hope you find them interesting and useful. I think "xor" would be logically equivalent to the statement using "or" since it cannot be the case that {u}={x} and {u}={x,y} both hold if u ~= v because then u=x=y and you derive the contradiction u=v from using either of the two propositions {u,v} = {x} or {u,v}={x,y}. I just didn't happen to think of using that logical operator.
We are trying to define an ordered pair [x,y] as the set {x,y}. But that doesn't work, because [x,y] and [y,x] would have been the same set. You see further in the video that the usual definition of an ordered pair is {{x}, {x,y}}; this definition has an advantage that it works even in absence of axiom of regularity.
+João Vitor Denardi Bosa Thanks for the compliment. That theorem, at least in my mind, just tells you that if you have some set, A, then the construction of taking two power sets will give you all the possible ordered pairs between two elements in A. I'm not sure if it has any practical use in a computer science or programming sense.
What do you mean by a*a, the same as AxA in the Video? And no, the Powerset of the Powerset of A, your pp(a) is not AxA. Simple because Powersets has no order. But it can contain ordered pairs als elements. Big difference. And: (x,y)e a*a | (x,y)e pp(aUb) Why do you ned a B here? of course you can write (x,y)e pp(aUbUcUd and so on), even if B, C, D are empty. Its enough to show that (x,y) e pp(a). (still learning)
You can define an ordered triplet [x,y,z] as the pair [[x,y], z]. I think the set {{x}, {x,y}, {x,y,z}} would have worked as well. Another way could be to define it as a function from the set {0,1,2} to {x,y,z}: {[0,x], [1,y], [2,z]} (once we define a function as a set of ordered pairs, and a natural number as a set of all numbers less than itself - 0 is the empty set, 1 is {0}, 2 is {0,1}, 3 is {0,1,2}, and so on).
+edwardjiang7 Thanks, I'm a bit new in making videos, but would have have any recommendations for better screen recording software (I currently use CamStudio)? The audio probably sucks in my older videos but I think it's better in the newer ones.
mdphdguy1 A good screen recording is OBS (Open Broadcast Software). It's free and open source. It does take some time to set up but UA-cam is a great place for tutorials! Good luck!
The only place where I found a clear breakdown of the definition of ordered pairs using sets. I couldn't wrap my head around the fact 'why is the definition like = {{x}, {x,y}} and not = {{x},{y}}, but this video showed me why. Thanks for this tutorial. Many regards.
Thanks to this video series I am beginning to understand why mathematicians like Frege, Russel and Hilbert thought that set theory is the foundational starting point from which all the other branches of mathematics can be built. Before viewing this video series I had only the high school version of set theory as Venn diagram.So I wondered how the notion of sets could be used to derive or construct mathematical objects like ordered pairs and coordinate plane. Now I have a intuitive sense of how the rigorous definitions of an ordered pair and Cartesian product will lead to talking about functions and relations. It's hard to follow philosophical issues in mathematics without doing some actual math yourself, and this video series definitely makes an intimidating subject quite easy to understand. Part 3 was a tougher going than the previous two, but it was definitely worth it. Thanks!
John Ok Glad you enjoyed the video. One thing I really tried to do in this video was give a clear explanation of the proof that = iff x=u & y=v. I'm not sure if I succeeded but it's important to understand in order to see that this description of ordered pairs using sets actually works in preserving the "ordering" property we want. By the way, ordered triples have a similar description, where (an ordered triple) is first represented as (an ordered pair) which can then, of course, be represented as a set, as can any other ordered pair. Then, you can abstract it to n-tuples (n is an integer) by nesting them in lower order tuples.
mdphdguy1 Representing n-tuples by nesting it down to an ordered pair, which can then be represented as a set - now that's clever!
Very helpful video! This series has been changing set theory from vague concepts that I largely just memorize into a more logical and intuitive way of thinking. Awesome!
I've never seen the set theoretical justification for ordered pairs. It really makes a lot of sense and helps me to understand the construction of coordinate systems; so that's awesome. And that diagram at the end was freaking sweet.
Combine this with metric spaces and you get something way cooler: linear algebra.
Great pace and style by the way. Super easy to follow. Very refreshing.
Thanx a lot for those lectures on set theory . I enjoyed every second , this is how mathematics should be explained . Keep the good work up . Many thanx .
wow, this was the best explanation of the kuratowski ordered pair definition Ive seen. I mean, most proofs with this def usually follow closely the same lines you did, but Ive never seen a derivation in which where you created that conjunction statement explicitly before.It was elegant because once youve eliminated the innermost two cases, then the two outermost automatically follow by AND.
Btw, just found this series, and I'm looking forward to it. Been reading Enderton's set theory and Ian Stewart/David Tall's Foundations, and I have a real hard time grasping the proof of the Recursion Theorem.
12:42 that should be an xor, no?
what is a powerset of a powerset? How can I find out more about it? is it the same as a generalized product of a set function?
Hello, sir. Do you presume that most of your practice problems have solutions online? It would be nice if you had a video in which you went through them all in sequence, although I know what a burden that would be.
+Mitchell Acampora
It's a tradition in higher mathematics to propose practice problems and never give you the answers. But if you were curious about any of them, please ask.
Mathoma, it seems to me that the benefits of such "tradition" don't warrant the risk of perfecting a flawed approach to attacking a problem. At any rate, given the nature of the medium via which you present your lectures such information is much more conducive to instilling confidence in your audience and reaffirming their grasp on the subject matter, no?
+Mitchell Acampora
I was just saying that tongue-in-cheek. Do you have a question on one of the problems?
Out of curiosity, why isnt "xor" used in the proof @12:46? Thank you for the videos!
+Samuel Cramer You're welcome and I hope you find them interesting and useful. I think "xor" would be logically equivalent to the statement using "or" since it cannot be the case that {u}={x} and {u}={x,y} both hold if u ~= v because then u=x=y and you derive the contradiction u=v from using either of the two propositions {u,v} = {x} or {u,v}={x,y}. I just didn't happen to think of using that logical operator.
I'm confused about 4:41. Where does the property {y, x} = come from?
We are trying to define an ordered pair [x,y] as the set {x,y}. But that doesn't work, because [x,y] and [y,x] would have been the same set. You see further in the video that the usual definition of an ordered pair is {{x}, {x,y}}; this definition has an advantage that it works even in absence of axiom of regularity.
what a helpful video I was having an assignment which I don't even know where to start but thanx very much for dis video LIKE LIKE
hi is it possible if you can send a proof of the excercise thanks
Thanks for the video!
Is the last theorem ( belongs to the power power set A ) useful for something?
+João Vitor Denardi Bosa Thanks for the compliment. That theorem, at least in my mind, just tells you that if you have some set, A, then the construction of taking two power sets will give you all the possible ordered pairs between two elements in A. I'm not sure if it has any practical use in a computer science or programming sense.
Good job. Thanks!
Hey sir, if a*a=pp(a) then pp(aUb)=(aUb)*(aUb)? Then wouldn't it be possible to have (x,y)e a*a | (x,y)e pp(aUb)? Thank you for the awesome content!
What do you mean by a*a, the same as AxA in the Video? And no, the Powerset of the Powerset of A, your pp(a) is not AxA. Simple because Powersets has no order. But it can contain ordered pairs als elements. Big difference.
And: (x,y)e a*a | (x,y)e pp(aUb)
Why do you ned a B here? of course you can write (x,y)e pp(aUbUcUd and so on), even if B, C, D are empty. Its enough to show that (x,y) e pp(a). (still learning)
@@RickB500 I don't even know anymore it's been awhile lol, I picked up programming since haha
For triplets, would we define:
:=
Or
:= {x, {x,y}, {{x,y,z}}}
Or
:= {x, {x,y}, {x, {x,y}, {{x,y,z}}}}
?
You can define an ordered triplet [x,y,z] as the pair [[x,y], z]. I think the set {{x}, {x,y}, {x,y,z}} would have worked as well. Another way could be to define it as a function from the set {0,1,2} to {x,y,z}: {[0,x], [1,y], [2,z]} (once we define a function as a set of ordered pairs, and a natural number as a set of all numbers less than itself - 0 is the empty set, 1 is {0}, 2 is {0,1}, 3 is {0,1,2}, and so on).
Very good video. It would be nice if it was better quality, maybe using a screen recording software. Otherwise, it's a great video! :)
+edwardjiang7 Thanks, I'm a bit new in making videos, but would have have any recommendations for better screen recording software (I currently use CamStudio)? The audio probably sucks in my older videos but I think it's better in the newer ones.
mdphdguy1
A good screen recording is OBS (Open Broadcast Software). It's free and open source. It does take some time to set up but UA-cam is a great place for tutorials! Good luck!
thanks