Proof Symbols Used in Math
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- čas přidán 25. 06. 2024
- Hello there! Have you ever seen crazy symbols while reading math proofs? First of all, I guess not a lot of people just read math proofs for fun. Anyways, today's video is all about the most commonly used proof symbols in mathematics. Writing proofs and reading proofs can be a bit difficult because it's very "up to the author's discretion" on what steps they want to include. Some people like to spell out each and every step while others like to write proofs that skip some more intuitive steps. In this video, I will talk about the different symbols and what they mean. I then go over a "translation" of a sentence written with the proof symbols.
Remember the other video I did previously on the squareroot of 2 being irrational via Proof by Contradiction. I go over that same proof but this time I write it with the math proof symbols. Some people like more english words while some people like more symbols. There's a bit of freedom when writing proofs but it can make it hard to understand and follow when there are only symbols. If you haven't seen my previous video, you can check it out here! • Proof by Contradiction...
Another set of symbols that you will see are certain letters that look bolded with double lines. These guys are apparently called blackboard bold, I guess that's the name of the font. These letters represent different number sets. I made a video a while back on these number sets so watch it here for a refresher! • Number Sets - Counting...
Sections in this video:
0:00 Introduction to Proof Symbols Used in Math
2:08 Meaning of the Proof Symbols
7:21 Translation of a Sentence that uses Proof Symbols
12:53 Squareroot of 2 is irrational (Proof by Contradiction)
19:28 Outro - Recap and Support the channel
Thanks for watching! Please drop a like and subscribe to my channel. Don’t forget to let me know if you have any questions in the comments below.
For more math videos like this, be sure to subscribe to my channel / @cavemanchangalgebrate...
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As a mathematician, I have a couple of remarks.
I've never seen the single line arrow --> used to denote implication. In every paper I've seen, double arrows are exclusively used for that shorthand. Single line arrows are instead reserved for functional notation, like "f: A --> B" to denote "f is a function from A to B," or for limiting behavior, like "f(x) --> 0 as x--> 0."
I'm surprised that you didn't mention the subset symbol. We use that all the time.
When we use \in in our written proofs, we always symbolically define the set on the right hand side. So we wouldn't use "Even #," but rather, say, 2Z.
We also don't use symbols as often as a first-year proofs class would have you believe. I'm much more likely to say "For every" instead of the \forall symbol, and "There is a unique" instead of "\exists !". These symbols are used more commonly on blackboards and in lectures, to get the idea out more quickly.
I will use the implication symbol ==> sometimes, but I use it as a replacement for the word "implies." And when I do this, it's usually in an effort to state the implication without explicitly committing to the hypothesis. I might say something like, "P ==> Q, which trivially implies what we want, so we may assume that P is false." I also only do this when P and Q are already symbolically heavy statements, e.g., "x \in Z ==> x^2 \in Z."
Contradiction symbols aren't really present in papers, but on blackboards, I've taken to writing ==>
Note that some of these symbols are precisely defined mathematical objects (like the Implication), but other symbols (such that, therefore) are semantical values that exist to make the proof readable for a human, but they are not themselves mathematical objects.
exactly
as a romanian i couldn't for the life of me figure out what s.t. meant before he actually said what it means
in romania this same notation is called a.î. meaning "astfel încât" which translates to "such that"
@@greengreen110 dude nobody cares about your country
they look so much more complicated than they really are
"Abnormally simple" I love that phrase.
The two symbols for implies are similar hut have sifferent meanings. One on the left is "proven to imply" the other is "claimed to imply".
also you use the left one for statements and use the right one for formulas
There's really not enough established convention to distinguish between these two senses. You will need to clarify with the author. As with Xracess, I know of a context (in the Isabelle/HOL proof assistant system) where => is used at in the context of theorems implying other theorems, and -> is used as a binary operation in logical formulae.
Both "⇒" and "→" are used for implies in logic. Although the first one is more common, there usually is not a distinction between the two.
Thank you for this video, I found it really helpful and enjoyed it!
fact: .'. is therefore
and '.' is because
Excellent video. Thank you!
I've seen the contradiction symbol written as two arrows touching point-to-point (like this: -->
Thanks @ki for sharing! I have seen the two arrows touching point-to-point before but not the diagonal hashtag symbol. I'll include this on a future video coming out next week on the topic of Proof by Contrapositive.
ST can also mean Subject To. It pops up in optimization problems. For instance maximize some function subject to the sum of the independent variables equals X.
I NEEDED THIS VIDEO SO MUCH
Upside-down T means this is a contradiction, and regular T means something like "this is a true / logical statement".
I like the upside-down triangle of dots, which mean "because".
I've also seen contradiction by ->
Upside down T means "perpendicular" in my mathematical dialect.
@@ExplosiveBrohoof That too, in geometry.
Very good video, thanks!
@17:12 some people use
Amazingly clear. Thank you so much!
This is an amazing video
One symbol my Math teacher loves using is : before =.
:= stands for "is defined as"
Say, for example, the definition of a function f
f : |R* -> |R
x |-> y := f(x) := (x²+1)/x
Or, to be more succinct
f : |R* -> |R
x |-> y := (x²+1)/x
Or
f : |R* -> |R
x |-> (x²+1)/x
Huh, I've been using it to mean 'is reassigned to' for iterative stuff. I suppose I really should just be subscripting.
@@pauld9690 "≔" is commonly used for assignment in algorithmic contexts, so its fine to use it that way
Missed "QED", or square symbol.
I prefer the colon : for s.t.
sir respect from J&K
Great video.
Thank you so much
By the way sometimes ':' is used for such that.
really good video, love u
Ur very good .. U helped me ur awesome
hello teacher
i really love math
but i feel its hard
so i want to learn from zero
do you have the first lesson of math?
Another common one is iff. which stands for "if and only if". iff. and bidirection implication () are logically equal.
0:20 often in math "symbols are used to do some operations"
So too in truth functional logic sirm Even in the predicate calculus, the implication arrow means to do a calculation (unless they are in strings transformed by rewrite rules.)
And that arrow does not tightly cohere with the english 'implies'.
Whilst the meaning is different to that of the word 'implies' in regular english, i believe A⇒B is spoken as A implies B, and ⇒ is referred to as the 'implies sign', even if it doesn't really mean 'implies' in the usual sense.
@@josephthomas4900 Indeed. I believe logicians call ot "material implication". It means simply "(not A) or B".
Which is not to say other logics are impossible. But this is standard.
Therefore? But there are only three
I was under the impression that the double-headed arrow meant "if and only if".
It does. "If and only if" means the same thing as "implies both ways."
Has anyone ever written a comprehensive dictionary or encyclopedia of mathematical symbols.
The closest thing to this is the book "Principia Mathematica"
↯ Is the symbol I've seen used for contradiction most of the time.
0:26 GG
He'll I am student from India
symbols plural
Pretty much nobody uses the "three dot triangle" symbol. I have a phd in math, and I've never ever seen anyone use that symbol.
It's usage tends to be pretty specific, it's often the sort of thing where you'll prove a lemma, then have the logic after it where you can reference already showing the lemma(or cite it from elsewhere) therefore step where we use the lemma without going through the intermediary steps.
So it's very much a "step 1, step 2, invoke lemma therefore step 6, step 7, step 8" style progress. For the most part it's not nice, it's better to actually use the steps from the lemma/invoked proof if it's not well known rather than citing it, but it does get used.
My junior high algebra teacher used it, so I keep it handy for my own notes.
I've seen it a shit ton, maybe one of our experiences are just skewed
very common in engineering
f(x)!?!?!?
There existS a word which means what you're trying to say.
"There, exist.", is a command, of sorts, but, "There exist symbols", is nonsense, because "exist" is a verb, with no subject, but "exists" is a quality, kinda like a preposition.
I am not a professional English teacher.
What is the point of standing in front of written mathematical narrative and explaining it to baffled learners? Use overhead projector or long pointer to explain away finer points of ready solutions.
💚 Super nerd! 😅💚
You are confusing symbols that clearly don't have the same exact meaning. Beware of the *types* .
This was super important and really well done -- except for your self-deprecation, which was not funny and a total waste of time. You're a teacher! Get on with it!
This is the proof that mathematicians are lazy af😂
And don’t take me wrong, being lazy while keeping the work effective (in this case communication) is great and it is the core principle that drives us to a better world
✓°→√Ω¶{×÷}[]≤≥⟩⟨%±-·ⁿ⅒
Why not just write words. I don like when people overcomplicate with maths for no reason at all.
Like mfs who "prove 1 + 1 = 2"
He touched on this in the video, but there are two main reasons:
First, it’s much quicker to write. It might be hard to appreciate this if you’re someone who doesn’t write proofs often, but having to say “for any real numbers a and b there exists a third real number c” over and over again is super annoying and cumbersome compared to “∀ a, b ∈ ℝ ∃ c ∈ ℝ”. Again, writing it out once is reasonable, but if you need to say the same thing three times in a proof, and do five or so proofs in a homework assignment, you very quickly come to appreciate how these symbols shorten things.
The second reason is that unlike English words, these symbols correspond directly to logical concepts. My favorite example of this is the word “is”, which is used in English to mean equality (the king of Camelot is Arthur Pendragon) but also sometimes to mean element inclusion (Arthur Pendragon is a king) and also sometimes to mean set inclusion (a king is a monarch). Mathematical notation lets you distinguish between these: (Arthur Pendragon) = (king of Camelot) is the first statement, (Arthur Pendragon) ∈ King is the second statement, and King ⊆ Monarch is the third statement. Different meanings get different symbols so there is no possibility for confusion.
graphic thumbnail, graphic logo; but video on a physical board - no thanks.
Would you like your money back?
What's wrong with that?
Don't forget to wash your hands on the way out. Bye.
Fr was a bit disapointed
I know right man! I mean like, every other thumbnail on this website is a FLAWLESS representation of the video that they're on!
You talk too much , you flap your hands you could take off. You should have prepared your talk so your words are minimal . A boring explanation of set theory notation with some silly comments thrown in for good measure. Sorry to be brutal but if you are going to make videos you need to be clear and not tell the audience they know when clearly it is your role to tell them. You decided to be the teacher.
Thanks 👍👍👍
p and q are co-prime. I would express it as "GCF(p, q) = 1"
⊥⊥⊥ could also represent a contradiction