Actual Proof 1+1=2
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This video presents a clear and concise proof of why 1+1 equals 2, a fundamental concept in mathematics. It breaks down the logic and reasoning behind this basic equation, making it understandable for anyone interested in math.
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First
Actually we can prove 1+1=2 by proving lim x->1 x+1=2, using delta epsilon
You used addition in your definition of addition
And a bit more in, a giga condensed form, on my website
@@krzysztofszczepanik8380true
That hit hard when he said "If you really want to up your math game..." at the end of the video where he explains why 1 + 1 = 2.
What do you mean? Are you being ironic? I hope so, because this Bri, so called "The Math Guy", is the most intellectually dish0n3st math youtuber out there.
Bri's Mozart!
Hit hard? Because is s1lly af, right?
Meanwhile john hush proving 1=2 in every single way possible 🗿
That looks like a rather immediate task: there are no such ways
@@ilmionomenonlosothey’re all division by 0
@@ayuballena8217 of course they are…
@@tri99er_ exept…
@@ayuballena8217 except if we assume that every solid in 3 (or more) dimentions are mesureable :)
0:20 at this point the 2 circles and the loading icon came together perfectly lol
upd: maybe only in my phone
Can you explain the set theory behind the definition of addition?
You can define the successor of a set by adding {set} to it. So you have a set X that represents some number in Arithmetic. The successor will be {X, {X}} where {X} is the only set whose unique member is the set X and X and {X} cannot be equal by the foundation axiom of Set Theory.
Imagine writing this all down while solving a complicated math problem.
It's funny it's actually common sense if you think about it. Another way we could write the equation is more expansive. A=N B=M and C=B-1. Then writing A+B=1+(A+(B-1)) that plus one moves a few spots but if you can sub S for one you will always get that the left and right sides are equal.
I should clarify it is simplistic the way you explain it however it is still a strange concept if not approached right.
How and why are you using the + in the definition of +? S(a + c) includes a + but it is in the definition of +
Do you just need recursion?
@@fatfurry Yeah, you use recursion until you have a + S(0)
@@ViktorLoR_MainuDo you just need recursion?
@@mcr9822 Yes, you use recursion until you have a + S(0)
@@ianmccurdy1223 Do you just need recursion?
This is some terrance Howard level stuff
Nat = Z : Nat | S : Nat -> Nat
1. a + Z = a
2. a + S(b) = S(a + b)
S(Z) + S(Z) =[#2]=> S(S(Z) + Z) =[congruence #1 over S]=> S(S(Z)) Q.E.D.
Actually its a window if you combine it all
most life-changing moment in my life.
Same
What about fields? 1+1 = 0 in a field of characteristic 2, or, 1+1 = 0 (mod 2).
1+1=10
I'm not sure if S(1+0) is equal to S(1), I need proof
He showed that, by definition, a+0 = a
We know by definition that "x+0=x" for an arbitrary "x".
So, we have that "1+0=1" is true, and by definition of "=" we have that given an arbitrary property "φ", then "φ[1+0] φ[1]" is tautology.
So, particularizing "φ[x]" as "1+1=S(x)" for an arbitrary "x", as he's already proven in the video that "φ[1+0]" is true, by modus ponens we have "φ[1]" is true, and therefore "1+1=S(1)" is true.
So, by transitivity and symmetry law, we have: "x=y" and "y=z" implies "x=z", and "x=y" implies "y=x" for "x", "y" and "z" arbitrary. Therefore, as we have "1+1=S(1+0)" implies "S(1+0)=1+1" and "1+1=S(1)" is true (as we've already proven), we conclude that "S(1+0)=S(1)". □
@@GabriTellNicely done!
@@GabriTellNicely done
Easy, S(S(0)+0) =[congruence of a+0=a over S]=> S(S(0))
1️+1️=11
Finally… the correct answer 🤪
Proof by definition
Okay but how a + b = a when b = 0? Do we need a proof for that
This is true by definition.
apple+ apple= 2 apples
change my mind!!
I think its sad, not that so many people didn't understand this excellent video, but that so many people in their pride and anti-academic hubris are here mocking in the comments something which they failed to understand with an undeserved sense of superiority.
Humility is as necessary for learning as intelligence and probably more so.
Making jokes doesn's necessarily mean mocking
@@almabatekert_villanykorte3387 Obviously.
I am not sure that you really proved it. You defined a "+" operator through the successor. Therefore you show that 1+1 defined that way is 2. But I think you would still need to show that the way you defined "+" all the other laws like commutativity and associativity still are valid for your "+" operator.
He proved that given that given
a + S(b) = S(a+b)
then
S(0) + S(0) = S(S(0)).
1️⃣ Adalah salah satunya dengan cara yang paling
Semuanya Berapa
I have spent the last 40 years of my life trying to disprove this postulate. I graciously accept defeat at your hands, sir. SIGH.
You just wasted 40 years of your life, doing nothing 😊
0(1+1)=0(2+1)
Cancel the zeros out!
@@LeoV6502 Illegal! I shall report you to the Math Police!
@@LeoV6502uhhh that’s not how it works
Mua ha ha ha ha haaah!!!
This is what the teacher expects you to do when they say "show your work"
I always enjoyed randomly inserting the phrase "and then a miracle occurred here..."
@@throughthoroughthought8064 lmao, i always did 90% of the stuff in my head, not even on tests just in schoolwork and the teacher would conplain
No, it's really not hard at all.
It was revealed to me in a dream
@@Allyfyn This comment was obviously a joke, in case you didn’t know that.
we got proof that 1+1=2 before gta 6
💀
This "logic" proof could be written in a computer language called Prolog:
_% Base case: Adding 0 to any number N results in N._
plus(0, N, N).
_% Recursive case: To add A and the successor of B, we first add A and B, then find the successor of the result._
plus(s(A), B, s(C)) :- plus(A, B, C).
?- plus(s(0), s(0), s(s(0))).
And that is basically how arithmetic is defined in logic programming
And here's a similar thing but for the programming language and proof assistant, Idris:
data Nat : Type where
Z : Nat
S : Nat -> Nat
one, two : Nat
one = S Z
two = S one
plus : (a : Nat) -> (b : Nat) -> Nat
plus a Z = a
plus a (S b') = S (plus a b')
data Equals : (a : ty) -> (b : ty) -> Type where
EqualityIsReflexive : (x : ty) -> Equals x x
theProof : Equals (plus one one) two
theProof = EqualityIsReflexive two
This doesn't need any sophisticated reasoning because `plus one one` simply evaluates to `S (S Z)` and so is already the same thing. The type | proof checker can verify it mechanically.
Also, if I were to use the standard library just these two lines would do the trick:
theProof : 1 + 1 = 2
theProof = Refl
@@mskiptr Excellent. Yes. Question Idris or Agda?
@@gheffz To be honest I don't have much opinion on Agda because I haven't really tried it (yet). It seems to have a bit more active community around it and I would guess the tooling should be less janky. Also, from what I've seen it's more focused on maths while I'm overall more interested in applying type theory to CS concepts. Oh, and then there are the fancy symbols used everywhere. I kinda like the idea but it has a massive potential to be really annoying and ugly.
@@gheffz Idris for the programmers, Agda for the scientists :)
The name of the video is miseading, in my opinion. This is not really a proof that 1+1=2, but a proof that the definition of addition presented here is consistent with the result "1+1=2". If it wasn't consistent, the conclusion had not been that 1+1 does not equal 2, but that our definition for addition is faulty. This kind of proof put the result as something we want to achieve, because we assume it is true intuitively but lack the formal tools to describe it: we try to build a system of axioms and definitions that will lead to the result we wanted it to lead, and than we prove we succeded. Therefore it is meaningless to say we "proved that 1+1=2". What we did was finding an extremely elegant definition for addition that is consistent with the intuitive idea we already had and just wanted a formal set of axioms that will lead to it. This definition, or an extension of it, might become interesting and useful when it comes to adding things we don't have intuition about, such as infinity.
Agreed. Axioms are supposed to be the basic self-evident truths, but there is no way that Peano axioms are more self-evident than the fact that 1+1=2.
The video proved that "1+1=2 under the Peano axioms", and mentioned that the Peano axioms give us natural numbers, that are identical to the natural numbers we intuitively know. Thus 1+1=2. Proving the equivalency of the two groups would be a different proof.
@@b3kstudio This video heavily implies that the Peano axioms are in some sense more fundamental that basic properties of natural numbers, without really explaining why. That's why it has left me a bit frustrated.
@@catbertsis I think it depends on how you look at it. You could argue, that there is nothing more fundamental than taking a pile of one sticks, and another pile of one sticks, and putting them together. But that happens to be the same as the Peano axioms. Which one is more fundamental? Piles of sticks, for which you need sticks, which are arguably quite complicated by themselves, or 4 laws that can be used to represent the same thing.
In the end, I think saying "this pile is 1, and this pile is 2" then taking two piles of 1 and putting them together is not a worse proof in any way, than the one in the video. Because really, 1+1=2, because we defined 1 and 2 that way.
@@b3kstudio formally. you are right, of course. But the motivation of such a proof is not to prove that 1+1=2, but to prove that Peano Axioms are worth something and not leading to results we don't want. From that point of view, "1+1=2" was the result we wanted to have and the Peano Axioms were the axioms needed to achieve that - they are the starting point from the point of view of Logic, but not from the point of you of what we really wanted to find.
My Second-grader says "nu-uh." And I can't disagree with them.
1:40 how can we use addition to define addition? How can that make sense?
look up what an inductive proof looks like.
He's not using addition to define itself, he's using the base case over and over again as a function of counting to define addition.
It's as said before an Inductive definition such as for the factorial: if a (different) 0, then a!=(a-1)! x a ,Else (a=0) a!=1. You don't use the operation to define it. You define it by an operation that you already know (multiplication for the factorial,successor for the addition) on the previous term. And by having defined the first term, you have define Ur operation for every terme, since the first term is always reach
You just repeat applying the definition if it’s not the base case until you get to the base case, where there is no such holdup.
Addition itself is an axiom, so it is not being used to define itself. It is just being defined.
We do not define the abstract essence of 'addition', but rather each specific case. Each case makes sense because it is reduced to the basic one, which is defined as true.
0 is very often included as a Natural Number when you are using set theory as the underlying basis since the Naturals are then defined as being the set of all possible finite cardinalities, and since the cardinality of the Empty Set is 0 that makes it a Natural Number.
Where 0 isn’t usually included as a Natural Number is when you’re working in Number Theory since 0 is an annoying exception in a lot of theorems involving factorization. It’s simply more convenient to define the Naturals as starting at 1 in that context so you don’t have to keep dealing with 0 as a special case.
You define addition with addition?? how is that proof bruh.
I'm with you, he makes no sense
It's a recursive definition. It wouldn't make sense without the base case a+0 = a. If you think about it and look for it on the internet, you'll see that it was a great choice to define things that way since you can prove a lot of things with the induction principle
If I give you a iPhone at 7:39 and another iPhone at 7:39 how many iPhones would I have given you?
It will eventually go to another definition
Recursion
So 0 or 1 being the initial natural is actually a big split in conventions, in ny experience German speaking areas were more likely to start with 1, French and English speaking with 0, though English was the most mixed of the bunch
And this split goes back even before Peano pubished his formalization, he was actually beaten to the punch by Dedekind (his formalization is equivalent, but also harder to state and closer to second order logic than first order)
In Dedekind's initial manuscripts he started at 0, but somewhere in the process he began starting it at 1, he never wrote down why he changed it, but if I were to guess, the way he was approaching proofs became more elegant and simpler to write after the change, others went with 0 because their approaches had the opposite side for elegance
0 indeed is not always considered a natural number, but it can be included, if needed. Also this way of addition extends to every whole number, not just naturals, so it is accepted here for convenience.
I love this proof so much! X3
Gonna become a preschool teacher to show the kids this thing so they are even more terrified for the next 10 years. :3
Wish these videos were out 12 years ago. Been wishing for an understandable explanation since high school, thank you so much!!!
I call this many cats 🐈⬛ one I call this many cats 🐈⬛ 🐈⬛ two if a put one cat next to one cat 🐈⬛ 🐈⬛ I now have 2 cats. Proved. Go home people
We invented 1+1=2 to describe one apple and one apple equals two apple. And now we go full circle to prove what we define
That's true, but some mathematical experts disagree with us .
Yes but can you prove that 6 was in fact scared of 7?
It is interesting how often in science and math the most tangible things are often the hardest to define in abstract terms.
I’m no mathematician, but I don’t think this is possible. Math is a form of representation. Assigning numbers to things is a uniform way to represent otherwise dissimilar and discrete things with one symbol. If I have a sheet of paper and a cup on my desk, I say there are two things on the desk not because the objects share some material property, but because I choose to represent them abstractly. This eliminates the need for me to be able to perceive a difference in the material objects, so long as I reason that they are indeed not the same object. So if I individually call those objects ‘one,’ I can declare that one ‘one’ and another ‘one’ makes ‘two.’ This verdict of nomenclature isn’t a mathematical law-it’s arbitrary. There’s nothing about the cup that makes it ‘one.’ In other words, the logic that outlines the definition of a number and affirms that 1+1=2 isn’t axiomatic. The validity of the statement is implicitly asserted to make mathematics’ epistemological laws possible, which all ultimately collapse when tasked to explain themselves, much like the paradox of trying to explain language using language.
if only youtube existed when Russell tried to proof the same thing! ;)
I am still waiting for a math video on TH-cam titled "The man who ALMOST broke math"
I need proof that this proof is legitimate before coming to the conclusion that 1+1=2
It took 379 pages to prove this in Principia Mathematica.
This is because the concepts weren't well understood. Nowadays it's much simpler. Because we have solid foundations for both Peano Arithmetic and models in Set Theory (so whatever method you choose to prove it with is straightforward).
it didn't take 400 pages to prove 1 + 1 = 2; the authors used that space to establish an entire new foundation of mathematics notation and definitions, and then they just happened to prove 1 + 1 = 2
I thought they didn't quite finish it.
Prove "1" = "one" and that "2" = "two."
For that matter, prove "1" or "one" = the concept to what they refer. You can't. These things are only true because we stipulated them as such. We could just as easily have stilulated the numeral "1" to correspond to the word "eight," and for both of those to correspond to the idea of 962.
In other math 1+1=0 bro 😂 this math in namex planet 😂
"you 'probably' agree with me"
well, probably is a little like maybe.
I'm not a mathematician, but you use an addition in the proof of adding?! S(1+0). Is that valid?
1+1=2 is not self-evident. Because it´s not always the case that one thing plus another thing just equals two of them. See, one idiot is just an idiot. But one idiot plus another idiot does not just give you two idiots, it gives you a problem.
You know, if you don't pay a university $100.000 dollars, you can just prove 1+1=2 by adding 1+1 together.
For a moment i thought you were going to talk abt the few hundred paged proof but at least this was a little more meaningful to think about.
This is essentially the same proof. For the one that takes hundreds of pages, most of the pages are used to define the rest of the Peano axioms that were mentioned in the video, so it's not actually hundreds of pages just to do 1 + 1 = 2. For that you only need successor and addition.
@@ZachAttack6089 The other Peano axioms are just that the successor function is injective and the axiom of induction. How do you need hundreds of pages for that? I think if one really writes hundreds of pages to prove that 1+1 = 2 that's not a proof that's an exercise in obfuscation.
It's not my purpose to sound like a complete rube, but speaking as a non-mathematician but someone who does appreciate mathematical truth and exactness, I think this sort of thing is a lot of nonsense. Imagine an ancient farmer with a goat, one goat. Now a second farmer brings the first farmer another goat, one goat. They put the second goat in the pen with the first goat and count and observe, "Hey, there are one, two goats. When we combine one goat plus one goat, we end up with two goats." Why must a proof that 1+1 = 2 be any more sophisticated than the observation of those farmers?
I'm just gonna copy and paste this comment cuz i think it answers ur question.
@davidmadar8894
13 days ago (edited)
The name of the video is miseading, in my opinion. This is not really a proof that 1+1=2, but a proof that the definition of addition presented here is consistent with the result "1+1=2". If the conclusion had been that 1+1 does not equal 2, than our definition for addition is faulty. This kind of proof put the result of 1 + 1 = 2 as something we want to achieve, because we assume it is true intuitively but lack the formal tools to describe it. we try to build a system of axioms and definitions that will lead to the result we wanted it to lead, and than we prove we succeded. Therefore it is meaningless to say we "proved that 1+1=2". What we did was find an extremely elegant definition for addition that is consistent with the intuitive idea we already had. This definition, or an extension of it, might become interesting and useful when it comes to adding things we don't have intuition about, such as infinity.
Terrence Howard does not like this video 🤣🤣🤣
Thats ridiculus you answered the question with question itself so why 1+0=1 dont need these weird things tp prove that you have fingers in one hand open 1 finger also in another hand and approach them so there is 2 fingers thats all
fingers are not proof
Before symbols (conventions) there were fingers.
Is this the rube goldberg machine of arithmetic?
You are an inspiration sir ✨🙇
you haven't shown that n = m implies S(n) = S(m) which you use when you say S(1+0)=S(1). You also haven't shown why 1≠0. you could use 1=S(0) and S(n)≠n but that hasn't been shown either
S(n) is a function which is by definition right-definite; that is the property you're looking for.
The second claim can't be proven from the axioms given in the video. The peano axioms have a statement for this: "zero isn't the successor of any number" or in first order logic, forall n: Nat, S(n) ≠ 0.
CGP Beige
Now do a video on the proof that 1+1=2 from Russell's and Whitehead's book, Principia Mathematica!😁
Zero is the absence of number. Fail
I’m dumber after watching this video than before it. Now I don’t even understand why 1 + 1 = 2.
You're gonna have to read Principia Mathematica
I think 1 plus 1 can be whatever number I want like 3, it matters not that it's wrong, but I'm offended if it's not therefore I'm right.
Let's be honest we live in an age when you can pick your own gender.
Facts and truth do not come into anything anymore. it's all about how you feel and not to hurt anybody.
I can't believe in this enlightened age with the internet etc we have become so dumb.
if you have one cookie, and i give you another cookie…
This doesn't prove anything, it's a tautology. The "successor" is the sum of that number and 1. And you define the sum as the successor of... which is the sum of... which is the successor...
the successor function is not the sum of that number plus one. rather, the natural numbers are defined in terms of the successor function. S(0) is defined as 1. S(1) is defined as 2. S(2) is defined as 3. S(78) is defined as 79. etc.
No, it's not "n+1", it's "first goes 0, then goes 1, then 2, then..." without any assumption about addition, equality, or other relations/operations. They don't exist until you define them explicitly.
One view + One view = Two views , simple enough
I am no mathematician but this proof feels circular. You're trying to prove addition works the way it does and then you use its definition to assume A+0 is equal to A in order to proceed with your recursive proof of addition.
Here's my proof for why 1 + 1 = 2. I don't even need language or name like one or two.
If I take a 100 lb sack of grain up to the third floor and then come back down and take another sack of grain up to the same spot then you can bet your life I'm going to try to find out what happened to the first one if I don't see it where I left it. I don't even need to have names like 1 or 2 or addition. It's just a real life experience expecting the result of both my actions to be there when I'm done. Addition, an operation on which the entire field of mathematics is built is simply a formalization of this real life experience by humans where they expect both objects to be there if they repeat the operation of moving objects.
Now imagine if real life was such that you would need to repeat a moving operation three times to get two objects (a built-in godly taxation system!) then I bet you that the addition operation in our mathematics would have been defined as 1+1 = 0 and 1+2=2 and 2+2=3 and the number axis would have been defined entirely differently with a different abstraction for natural numbers. And then we'd have youtube videos showing rigorous proofs on why 1+2=2 using the successor numbers or some other man-made concept.
It's called recursion, not circular logic and it's perfectly valid
math just make things up
this is why I love math👍
Marhematicians. How to complicate simple stuff.
But you explained hiw to do sums, with sums. U cant define somthing with itself.Its like saying the word "small", by defenision, is somthing small. U cant say that
Absolute nonsense
You can just keep feigning ignorance about proofs by saying “but what is ‘TRUE’”
And
“But what is “is””
Was taught growing up to always be skeptical and think critically. Wasnt sure about this one. Had to cover all my bases.
That's no proof. That's proving an axiom by using axioms that are based on this particular axiom.
o (1 circle) + o (1 circle) = o o (2 circles)
Edit: In hindsight, this is technically visual proof and is not reliable for calculations.
I thought it was by definition
It is, this proof is wrong. This proves that 1+1=S(1) but of course the last step S(1)=2 is by definition... So not a proof, just a definition.
@@gabrielbarrantes6946 I think you could theoretically prove it by using set theory…
Nah, that's ridiculous...
It basically says:
a+b = a + b - 1 + 1
Because
a + b = S(a + c) and b = S(c), so b = c + 1 and c = b - 1
Therefore
a + b = S(a + b - 1)
Which is just
a + b = a + b
They really try to make themselfs look smart by making obvious stuff weird....
After 53 years there are some things that I have learned are just true with ZERO interest in being proven correct. This is one. 🤣🤣
1+1=2 is more like a definition. You can come up with others that 1+1=3 as well.
question: if b=1 is the successor of 0 then wouldn't a=1 also be a successor of 0 the the equivalence of a+b would be S(0+0) which is 1. please help to explain
But why is S(1 + 0) = S(1)? How can you "proof" that 1 + 0 is 1? 0 is a real number and 1 can be real and natural. And now since we want the result of 0 + 1 we naturally going to use 1 as a real number. But after that we need to make it a S(1) which is a natural number. So how do we know 1 in real number is the same as 1 in natural number? I sound so stupid yapping like this😂
This proof is wrong. Addition is defined for all numbers. Not just naturals, or rationals, or reals. And it works the same for all types of numbers as well. And there is no discrete "successor" to anything except a natural number. You can't prove someone about the general case using a function defined only on a subset
This is the math equivalent of having to meet the minimum 1000-word count in your essay
WAIT! think 1 water droplet plus another water droplet is 1 water droplet
Shouldn't you have used Whole Numbers set? Whole Numbers is Natural Numbers set plus Zero
Bertrand Russell approves. Now, prove that 1 + 2 = 3
S(a+c)? But there is a 'plus' sign inside! It does not form a proper definition if you define something with itself.
i can't understand this video because you're assuming I know what words mean
imagine a universe where 1+1 is a hard problem and people have to study to understand it.
I don't understand why you'd need to "prove" this. Addition is an operator, a function. It is defined. It is defined as two when inputing 1 and 1. 1•1 -> 2
Thought it's by definition; i.e,. 2 is defined as being the successor of 1. What suggests thr operation isn't well defined?
everyone knows 1 + 1 = 10
we all need to get on the binary train
So now there are three ones?? This is madness.
The way you ended the video is sorta funny 😅 The way "If you really want to up your math game" just abruptly came in
ERrRrR but then how do you prove that? And how do you prove that? Proving math is just a goddamn paradox
Bahh… An axiom is something that has been agreed upon. An assumption. We could just as easily say that 1+1=2 is an axiom and be done.
This proof is beautiful. Thanks
Does this proof by describing subsequent numbers as successors still work if you are looking at real numbers rather than natural?
Youre not making watch this lul bro
Or just stick to the take one object and then take another of that, you'll have two. I'm convinced.
Yes but why S(1)=2 ? Because S(n)=n+1 ? But why ? And what is n+1 if we use S to define addition. And if u dont use addition, I feel like at the end of the day you use « intuition » (the one we avoided at the beginning) to define what is a Successor
That's how naturals numbers are defined via Peano axioms. So, 1 is defined as S(0), 2 is defined as S(1)=S(S(0)), 3 is defined as S(2), and so on. S is not "plus one", it's "next", so 0 is the first, 1 is next, then goes 2 etc. There is no intrinsic addition or multiplication, just a sequence of things. You can define an addition over them, the video shows you how. You can also define multiplication: a * 0 = 0; a * S(b) = a + (a * b). Funny enough, you don't even have equality out of the box. Let's define it: 0 == 0 = True; S(a) == S(b) = a == b; _ == _ = False
S(1)=2 because 2 is just a symbol that means nothing . I mean we can define set of natural numbers as N = {0, S(0), S(S(0)), S(S(S(0))), and so on …}. Numbers are just a symbols.
@@daniiltonkonog186 right, we can also talk about bijections between {Z,S(Z),S(S(Z)),S(S(S(Z))), ...}, {0,1,2,3,...}, {zero,one,two,three,...}, and other sets that enumerate natural numbers
bro at on video: "1+1=2" and on the next video: (9)^sin^2 x + (9)^cos^2 x =6💀
I recommend y all checking out type theory, you can actually implement addition in prog languages like agda
This could be generalized by adding the fact that 1+1=2 is e true iff it's belong to ordered ring(if a ≤ b then a + c ≤ b + c.
if 0 ≤ a and 0 ≤ b then 0 ≤ ab. In fact one can check by replacing usual addition with modulo addition and simply 1+1 mod 2=0 and so the statement is false) and it's multiplicative identity which is 1 and additive identity which is 0 don't coincide i.e 1=0 and 1+1=0+0=0 false otherwise its true and the ring which 0 and 1 belong to it it's trivial
just clicked this cause i’m getting into Godels therum even tho i don’t do any maths