The Empty Set & Vacuous Truth

[@BassySasskets]

I've just gotta say, I'm in an online course for computer science and my discrete math course was making no sense to me before your videos. Reading out of a book trying to teach yourself something never stacks up to learning from an enthusiastic, charismatic professor with a well thought out lesson plan. Thanks to you my grades are improving.

[@DrTrefor]

I'm so glad it's helping. Good luck this semester:)

[@ubuntukali2997]

You are not only the Best Teacher but also the best actor. You dramatized it in a more memorable way.

[@orchestra4841]

You rock!! I love your videos, you should talk to Sal in khan academy to try and get this course on there

[@NothingMaster]

Just because you own a basket, you’re not obliged to put anything in it. In fact the whole beauty of the idea of a basket is to allow you to put anything, everything, or nothing in it.

[@seanli6339]

great video man im taking discrete 243 online in a 6.5 week session and ur vids have been a lifesaver

[@derrickchronicles5627]

Thank You, so much the video was very informative. I'm studying Game Development for my Bachelor's in Science. Math is the key to success in my career. You are far better than any instructor making it easy for me to comprehend.

[@yunoletmehaveaname]

This was the best explanation I've heard of the empty set being a subset of every set.

[@kvartalskiy]

I think this Professor should have millions of views and likes.

[@rishabhnarula1999]

I think the empty set being a subset of every set is like a notion of the idea that if everything in the universe is seen just as number of things, then the number representing collection of smallest number of things would be there in the number representing any collection of all things. or another way of looking at it is if any collection of things is considered as a universe itself,then of course you could keep taking things away from it to make the collection smaller,and conclude this collection of things was inside what you originally had and conclude that nothing must be in it by the end of the process. hence nothing is in everything. One more idea is if it is not a subset of some set, then there must be something in it that is not there in the other set. there is nothing in it. hence the contradiction.

[@hououinkyoumaich2]

Now, I just feel smart.

[@petersisler1398]

If x in A, then x in B, or if x not in B, then x not in A is equivalent. In our case if x not in {1,2,3}, then x is surely not in { }, since { } is empty. Therefore { } is a subset of {1,2,3}.

[@petersisler1398]

Contrapositive Proof.

[@peters6591]

This video makes the point that the vacuous set can be included since it is empty in other sets. I would not have seen this simplicity without your aid.

[@thiwankarandeni509]

You explained very simply. Thanks! 💖💖

[@jordainegayle5563]

another way to look at it is, innocent until proven guilty. (Original state is ture until proven false) or even to look at it that B was at one point empty, which means B state was an empty set which if you add to another set would be an element, so it correct to say a set always contains the empty set as it's initial state, i may be wrong, i just started watching.

[@luisishere987]

Great way to put it. The way I look at it is that emptiness is technically always present in something.

[@ShadaeBalancesKnightAstro]

Thank you so much for taking the time to teach us. You’re absolutely awesome.

[@pimvankeulen]

Is the following statement true? ∀ x ∈ ∅ : 1 + 1 = 3 That is: “for all elements x in the empty set, we have that 1 + 1 = 3”

[@angelmendez-rivera351]

Yes, the statement is true, because there is no set for which 1 + 1 = 3 is true, and the statement is true for no sets, because the empty set has no elements.

[@MysticJabulon]

Yes. Because ∀ x ∈ ∅ : 1 + 1 = 3 really is ∀ x (x ∈ ∅ ⇒ 1 + 1 = 3), which reduces to ∀ x (F ⇒ F), and that in turn to ∀ x (T).

[@daretodo2256]

Very nicely explained dear ☺

[@Abuda7amHD]

If we say there is nothing in ∅, and nothing in this context is the absence of elements, wouldn't that mean that in order for ∅ to be a subset of a = {1,2,3}, a would have to include an absence of elements? a has elements which means it doesn't have an absence of elements, therefore making ∅ not a subset of a.

[@angelmendez-rivera351]

No, this is nonsensical. There is no such a thing as "an absence of elements." Absences are not things, so it makes no sense to talk about "containing them" as if they were elements. The precise definition of a subset A of a set B is that, if x is an element of A, then x is an element of B. Notice how this is an implication relation on the calculus of propositions. Since a false antecedent trivially makes the implication true, it means that "x is an element of {} implies x is an element of B" is true, so {} is a subset of B.

[@understandtheuniverse2199]

I too have this same doubt searching but no answers all are simply saying vacuosly true

[@angelmendez-rivera351]

@@CommonCensorship It is not a leap in logic. It literally IS logic. In propositional calculus, the truth table for the proposition A ==> B looks like {(T, T) |-> T, (F, T) |-> T, (T, F) |-> F, (F, F) |-> F}. In this case, we have A = "x is an element in {}" and B = "x is an element of S", where S is some set S. Now, A is false by default: {} contains no elements, so x is not actually an element of A. Since A is false, we have either (F, T) or (F, F) as the input (T stands for "true" and F for "false"). Both inputs have T (true) as the output. So with A being false, it does not matter if B is true or not: A ==> B is true, so by definition, {} is a subset of S.

[@angelmendez-rivera351]

@@CommonCensorship Also, every set has elements that are sets. So ultimately, every set in ZF set theory can be built from only the empty set. Sure, x itself is not an element of the empty set. But x can be an set that contains the empty set. For example, x = {{}}. Then the set {x} actually contains something. The natural numbers are just sets containing sets containing the empty set. For example, 0 = {}, 1 = {0} = {{}}, 2 = {0, 1} = {{}, {{}}}, 3 = {0, 1, 2} = {{}, {{}}, {{}, {{}}}}, etc. Once you build the natural numbers, you can build arithmetic, and then you can build the integers, the rational numbers, the algebraic numbers, the real numbers, and the complex numbers. This gives you algebra and analysis. Basically, with only the empty set, you can construct every set needed for doing all of the mathematics you learn from Kindergarten to 4th year of undergraduate school. Once you have that, you can really begin to build functional analysis and vector analysis, topology, differential geometry, tensor calculus, operator analysis, etc. You get all of this from only the empty set and the axioms of ZF set theory. With the NBG axioms, you get even more. Measure theory, Fourier analysis, graph theory, combinatorial game theory, order theory, category theory,... you name it. Of course, in practice, you would not go through the entire construction sequentially. You take the construction for granted in the beginning and you move on. So {} being a set is useful, logical, and natural. In fact, it is more natural and useful than starting from some element x that is not itself a set.

[@angelmendez-rivera351]

@@CommonCensorship There is no leap in logic. {} does have a well-defined value. This can be proven from the axiom schema of comprehension. 0 is just another symbol we use for {}. 0 = {} is true by definition. Similarly, 3 is just another symbol for {{}, {{}}, {{}, {{}}}}. The symbols do not represent different objects.

[@jdavi6241]

3:20 I make the claim that there is something in the empty set that isn't in the set {1,2,3}. that something is the lack of objects 1 2 and 3. What do you say to this?

[@AbidAli-bv2gl]

Excellent video, best course

[@ajsworld77]

Thank you for this, Trefor. Is the explanation you are giving for the difference between the empty set and the set that includes only the empty set sufficient to prove this? Naively, I would say yes - the difference is in the cardinality, the empty set has a cardinality of 0, while the set that contains only the empty set has a cardinality of 1. Or is there a better proof somewhere? Thank you!

[@stephenhemingway9435]

At 2:44, shouldn't the symbols read A ⊆ B?

[@DrTrefor]

Both symbols are commonly used - rather annoyingly - for subsets that allow the possibility of equality. It just depends on the text you are using.

[@stephenhemingway9435]

@@DrTrefor Thank you.

[@owlsmath]

Love this class! This makes it sooo easy to follow.

[@midshadow]

Thanks for the video! I have a question I was hoping you could shed light on. I was debating with a friend who said, "I wear my sunglasses.". He says this is vacuously true because he means "my sunglasses" are the objects in an empty set. I'm skeptical. My understanding is when speaking of empty sets to make vacuous truths you define them with universal or conditional statements. i.e. "I wear all my sunglasses.". Since this isn't a conditional, I'm assuming it is a universal statement. In math universal statements are denoted with the upside down A? When spoken I assume this means you use a universal quantifier like all, every, for each, nothing ect. With all being the empty box and my sunglasses being the objects in the empty box. Or you say, "I wear what's in the set of my sunglasses.". As I knew nothing on the topic when I said he made a false statement, I'm also skeptical of my understanding which makes my debating skills weak and he is extremely confident. Sorry if this wasted your time, heated debate over here lol

[@anentrepreneurknownasherma489]

Thank you for teaching me 😃

[@continnum_radhe-radhe]

Thank you so much sir 🔥🔥🔥

[@math_travel]

I like your explanation through the box. It seems that there would be an empty gift. you are so passionate. thanks~~

[@alfredkokou2013]

Vacuous truth should just be an axiom and not a logical truth. @ 3:18 what prevent one from arguing the other way round that if it is true then there must be "something " in the empty set that is also in the set B? Which leads to a paradox not true not false.

[@falayyou]

I'm rusty with math and trying to learn C++. what other math topics would you suggest brushing up on other than discrete maths and linear algebra?

[@glanced9684]

Hey man, could you maybe do a series about maths for ML?

[@namraaah271]

Thank you I really appreciate it

[@angithkrishna2939]

i have a small doubt so when you were giving an example if a empty set is equal to a set with numbers u used that rotated u but you didnt mention the line under it ,is it a mistake you made or it has a meaning

[@dawzrd2458]

2:37 I thought that was the symbol for a proper subset

[@gamerdio2503]

It's true though. A is not equal to B, so it's a proper subset.

[@SonuSingh-qw2bs]

Hello professor Trefor hope you doing well. I didn't confused how a expty set is a subset of a set with a objects because what everyone can see that it's a empty set then how a empty set is a subset of a set of objects how's this being possible when the defination is a set is a subset of another set only and only when every element of one set is also present in the second set then we can say that the first element is subset of another one. Please explain me?? Thankyou!

[@inoxide5454]

Ok Dr, riddle me this: If my brain has no cells in it (like it feels now), is it an empty set ?

[@tjkids_Official]

Yes 🙂

[@christianevans5471]

my mind stuttered when he said if it's false that it's not a subset, then it's true that it is a subset. Rephrased: if the statement that it's not a subset is false, then it's true. Would this be like a contrapositive or inverse?

[@lordapple177]

just because something is false, doesn't mean it's true. Hence the term vacuous truth which im pretty is kinda like a half truth.

[@user-np5cp4rj9q]

Does this mean A= {{}} is an element of any set ?

[@samridhiarora1951]

Hello..your videos are helping me a lot.. There's a request,, can you please also make videos on REAL ANALYSIS, I'm finding it kinda difficult and my semesters are about to come...so can you please make videos on it asap or can provide something to study from.. Please do reply. Thank you.

[@Juoa_F]

Why is there no horizontal line under the “C”?

[@ChaoticAwesomeSauce]

Tell me a Set is an array without telling me a set is an array. XD

[@angelmendez-rivera351]

There exists a set {} such that there exists no set x such that x is an element of {}. This is true by the Zermelo-Fraenkel axiom schema of specification, if the proposition φ(x) is the proposition that x is an element of U and x is not an element of U for some set U. A is a subset of B iff x is an element of A implies x is an element of B. Therefore, {} is a subset of B iff x is an element of {} implies x is an element of B. Since x is an element of {} is false, x is an element of {} implies x is an element of B is indeed true, because a false proposition implies any proposition, by the definition of implication in the calculus of propositions.

[@DrTrefor]

Well yes, I don't disagree mathematically, but also this is for students just learning about the basic concepts of set theory, so Zermelo-Fraenkel is something for quite a bit in the future:D

[@angelmendez-rivera351]

@@DrTrefor Ye. The reason I posted this was because the comments seemed to get hung up over the definitions of "subset" and the concept of vacuous truth. Resolving this with intuitive explanation would not have been easy for me, but providing the formal definition could still be helpful in clarifying the language and thus clearing misconceptions. After all, if people are failing to understand even the definitions, then the theorems are not going to make sense regardless of how you illustrate them.

[@backoffer3228]

Is {{ emptyNess }} subset of {1, 2, 3}?

[@gamerdio2503]

If you mean the set containing the empty set, then no. {1, 2, 3} does not contain the empty set. If it did, it would look like {1, 2, 3, {}}

[@backoffer3228]

@@gamerdio2503 yeah, that's exactly what I meant. Thank you for clarifying. How is it going, DIO GAMER?

[@gamerdio2503]

@@backoffer3228 It's going pretty good. How about yours... uh.... Mr. Moon Runes?

[@angelmendez-rivera351]

@@gamerdio2503 No, you are wrong. The empty set is a subset of {1, 2, 3}. In fact, the empty set is a subset of every set. He is not asking if the empty set is an element of {1, 2, 3}, but if it is a subset of {1, 2, 3}. Being a subset and being an element are defined differently.

[@gamerdio2503]

@@angelmendez-rivera351 I know that now, yeah. And jeez, I hate how I said that so confidently. Confidently incorrect, huh?

[@codatheseus5060]

I thought that empty set symbol was the null terminator, which just tells the reader to exit the set at that spot, regardless of where it is inside that set. Maybe that compsci s**t in my head isn't necessarily the correct way of looking at it

[@ummeaysha593]

THATS REALLY FUNNY WHEN I SEE THAT THERE IS A BAG AND NOTHING IN IT .BUT STILL IT IS SOMETHING.AND I CAN SAY THE THING I SEE IS AN EMPTY

[@Junker_1]

I have always great difficulty with this one. We know that in the second set we have three elements 1,2,3. We don't have an empty set in the second set although we could simply add one by adding the null set if we wish to. But it isn't stated. How could an empty set then be in the second set? It doesn't make sense. Imo an empty set means something. It is empty, no fictitious things are in it like a pink elephant and such no it is empty (you even show it as an empty box). To me, that means that we can prove that the empty set is not a subset of the second set. I have never got this. To me it is false. There was certainly in the empty set that isn't in the second set. And that something is nothing (empty), that is not the case in the second set which contains 1,2,3.

[@c7hu1hu]

Nxx ai no aikhoo-ke njeh aikho!

[@wrapperofficial8882]

ahm..what are you doing on x album?

[@bibasniba1832]

Thank you Trefor!

[@curtpiazza1688]

Ooooh! 😮

[@anentrepreneurknownasherma489]

I’m still going 🔥☮️ Very slow rate but I’m working to be more effective

[@hritikbansal14]

Helpful.. Thanks..

[@fxizzy1874]

Why are you on x album

[@sharoonaftab8894]

Eh what?

[@qandos-nour]

thank you

[@DrTrefor]

You're welcome!

[@rishabhnarula1999]

We say that an implication p --> q is vaccuously true if p is false. Since now it's impossible to have p true and q false. That is we can't check anymore whether the contrary, p being true and q being false,can be.Since p being true is non-existent. So we take the implication as true. For eg. If 3 squared = 27,then 2+2=5. Can we check if it is indeed true that 3 squared equals 27 then 2+2 is not 5. No. Because 3 squared equals 27 is non-existent. Or false. So we can't check if the statement is false. Hence it must be true.

[@abdulwaheedyarik4396]

Sir g plzz group theory ki book jonhn fraleigh ka lecture upload tu karo

[@natespears6443]

Wouldn't that mean the empty set contains an infinite number of dimensional empty sets?

[@Google_Censored_Commenter]

I see no good arguments why I should accept the concept of empty sets being valid. Anyone can give any? Since empty sets don't exist in reality, what's the purpose of working with them? No problem will ever present itself that requires them.

[@andrewharrison8436]

Empty sets happen all the time - every time you take the intersection of disjoint sets you have to deal with the empty set.

[@Google_Censored_Commenter]

@@andrewharrison8436 Obviously within the framework of set theory, where the assumption is needed, you can make an argument. I'm talking about a practical problem in reality, not in the fictional world of abstract maths.

[@21DPH045]

Let's say there's a set A={1,2,3,4} Here,empty set is a subset of A but empty set does not belong to A,how? Please do give me a reply!

[@MikeRosoftJH]

Distinguish the 'subset' and 'element' relation. Empty set is a subset of the set {1,2,3,4}, but it's not its element.

[@Leo-ig6yl]

I feel the set is somehow related to the JSON object🤔🤔

[@KoopavonRox]

Why are they messing with the simulation

[@stevenshrii]

Explain 'go set'

[@EvansCaldwellOwusu-Antwi]

please i just started , and I am not that good with math , can someone offer a hand .

[@lukatalev2639]

nicee

[@ollyop5762]

Your on xxxtentacions album “Bad vibes forever “ ❤️

[@someperson9052]

What?

[@EricSun777]

@@someperson9052 Yeah i dont get it either.

[@GoatzAreEpic]

The empty set represents the amount of girls who want to be my girlfriend :(

[@mohammaddamirchilu2221]

💌💌

[@graeme011]

The existence of a set depends entirely on the existence of its elements. No elements means no set. The empty set does not exist! Neither does a set containing the empty set! There's nothing there!

[@angelmendez-rivera351]

This is nonsense. The axiom schema of specification implies the existence of the empty set, so you are incorrect. A set with no elements is still a set. "No elements" is a well-defined property, because there are no non-existing elements in the set.

[@mkmilla4765]

Anyone here from xxxtentation's bad vibes forever??

[@wachowski9525]

what part of the album includes this video/guy lol

[@christianevans5471]

this may sound cringey; What if God is a vacuous truth, or however you wish to call some deity or higher power?

[@wernerhartl2069]

The empty set E is not a subset of A. Proof: No member of E is a member of A.

[@DrTrefor]

What is the precise definition of being a subset?

[@wernerhartl2069]

@@DrTrefor Every member of E is a member of A. There is no member of E which is a member of A, E can’t be a subset. On the other hand, E is a subset of A because there is no member of E which is not a member of A.

[@wernerhartl2069]

@@DrTrefor Thanks for reply by the way. I suggest Wiki article on vacuous truth: en.wikipedia.org/wiki/Vacuous_truth

[@angelmendez-rivera351]

The definition of vacuous truth actually disproves this statement, so indeed, {} is actually a subset of A.

[@wernerhartl2069]

@@angelmendez-rivera351 You can prove a vacuous (empty) statement either way. Is E a subset? Yes. Is E a subset? No. You can choose either one, but it’s not mathematics.