Universal and Existential Quantifiers, ∀ "For All" and ∃ "There Exists"
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- čas přidán 28. 05. 2017
- Statements with "for all" and "there exist" in them are called quantified statements. "For all", written with the symbol ∀, is called the Universal Quantifier and and "There Exists" , written with the symbol ∃, is called the Existential Quantifier. A quantifier turns a predicate such as "x is greater than 7" into a statement that can be true for false. For instance, "For all x, x is greater than 7" is false as 2 is not greater than 7, but "There Exists an x such that x is greater than 7" is true as 8 is greater than 7.
Learning Objectives
1) Be able to use the Universal and Existential quantifiers in a sentence
2) Observe that a Quantified Predicate is a Logical Statement
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TheBrownBear when some random video on youtube makes you understand more than when your teacher teaches you
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Your explantions and examples about the universal and existential quantifiers are so well.Thanks.
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Thank you thank you!! From reading the textbook, watching your videos, I was one of the few not lost in class last night! Prepping for next class trying to understand Quantifiers and you did it again! Huge thanks!!!
Nice! Glad it helped:)
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Can a “∀” substitute “+/-“ in a given tolerance? And will the answer still be within tolerance?
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Do you have a video of choice function and scope?
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Clearly explained. Thank you.
what is the universal quantifier for sentence "everyone saw someone"?
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@Trefor Bazett But how it can be both,I am precisely asking what does the definition of even numbers refers to.Or for simply,the
definitions of chairs,tables,spoons
etc refers to a class satisfying the stated property or these terms symbolise any object satisfying stated property.
2 is even number.True
But in what sense?
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Please do you have a full course tutorial videos for logic? Thank you
Trefor Bazett ok I’ll go through it. Thanks
I would like to ask if there is any difference between "every" and "all".. I think they express different meanings but can the universal quantifier express the difference?
Trefor Bazett Thank you very much😊
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Can you relate this to rayo number
Is it allowed that someone writes the predicate first then the quantified variable? For instance, P(x), ∀x ∈ D instead of ∀x ∈ D, P(x).
@@DrTrefor Yes, I can catch that. I found few books or academic modules use that kind of writing style, but honestly it's bothering when I read it.
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Your videos are so great! Many people from my online class agree. We were all wondering though, how do you film these videos lol. We're assuming you're not writing backwards behind a glass frame?
@@DrTrefor ohhhhh that makes sense to envision it that way LOL thanks so much, i'll let my classmates know in our discussion board that you're not writing backwards hahah :D
good lesson ....but how could you write in an opposite direction
i mean for all people in the world, there exists.....................
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Sir plzz help
All integers are rational.
Sir how to write logical notation of this statement..
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Thank you teacher.
Also, if you want to stack quantifiers, be very careful of order. For a good informal example from Discrete Mathematics with Applications by Susanna S Epp, if L(x, y) is a predicate meaning “x loves y”, then AxEy L(x, y) means “Everyone has someone they love” (more formally, “For all people x, there is a person y such that x loves y”), but EyAx L(x, y) means, to quote the book, “that there is one truly amazing individual who is loved by all people” (“There is a person y such that for all people x, x loves y”). This ambiguity also comes across in the more informal English wording “Everybody loves somebody.” This is only when mixing E and A types; when the quantifiers are the same types, order doesn’t matter (ExEy and EyEx are the same).
In fact, I was recommended this video by this online textbook thing called ZyBooks which I’m learning from, and provides a nice analogy for how the stacked quantifiers work. Basically, when evaluating a proposition with quantifiers, you can think of it as like a game between two players, E and A. These two fill in the variables in the proposition in order of the quantifiers; when A fills one in, it’s trying to make the proposition false, while E is trying to make it true.
For example, with a statement like AxEy (x+y=0), the first turn goes to A, who may fill in x with any number, and wishes to make a false proposition. However, no matter what number he fills it in with, E can then pick a y=-x, so x and y sum to 0. Thus E wins, and the proposition is true. However, with EyAx (x + y = 0), E goes first; no matter what they fill in y with, A is free to choose any value other than -y for x, which makes the proposition false; thus A wins and it is false.
For a slightly more complex example, try ExAy (abs(y + 1/y) > x). E goes first, and can fill in x. It turns out that if they pick any number less than 2, A is foiled; no matter what they put into y, y + 1/y never has an absolute value less than 2. As such, A loses and the proposition is true.
Too long
Thanks a looot buddy
The analogy of E (the existential quantifier, symbolized by ∃) and A (the universal quantifier, symbolized by ∀) as players in a game is a conceptual way to understand how quantifiers work in predicate logic.
Here's how the two "players" are defined in this context:
E (Existential Quantifier ∃): When ∃ precedes a variable, it means "there exists" or "there is at least one." The goal of the E player in this analogy is to find at least one instance that makes the statement true. E is satisfied with finding a single example that works.
A (Universal Quantifier ∀): When ∀ precedes a variable, it means "for all" or "every." The goal of the A player is to ensure that the statement is true for every possible instance. A wants the statement to hold universally, without exception.
When these quantifiers are "stacked," or placed in sequence, it creates a game-like scenario where E and A take turns "choosing" values for the variables they are quantifying, with their goals opposed. E tries to prove the statement true, while A tries to prove it false. The order of the quantifiers is crucial because it determines the order of play in this game.
Let's use your examples to explain the analogy:
AxEy (x + y = 0):
A's turn (Universal Quantifier ∀x): A must pick a value for x that it believes could lead to the proposition being false for some y.
E's turn (Existential Quantifier ∃y): After A picks x, E tries to find a value for y that makes x + y = 0 true.
Outcome: E can always pick y = -x to make the equation true, regardless of A's choice. Therefore, E wins, and the proposition is true for all x.
EyAx (x + y = 0):
E's turn (Existential Quantifier ∃y): E must choose a value for y hoping to make the proposition true for all x.
A's turn (Universal Quantifier ∀x): A will then try to find a value for x that does not satisfy the equation x + y = 0, aiming to prove the proposition false.
Outcome: No matter what E chooses for y, A can always choose an x that is not equal to -y, making the proposition false. Therefore, A wins, and the proposition is false because it does not hold for all x.
ExAy (abs(y + 1/y) > x):
E's turn (Existential Quantifier ∃x): E needs to pick an x such that no matter what y is chosen afterward, the inequality abs(y + 1/y) > x will be true.
A's turn (Universal Quantifier ∀y): A will then try to pick a value for y to disprove the inequality for the chosen x.
Outcome: If E chooses any x less than 2, A cannot find a y to make the inequality false, because the absolute value of y + 1/y is always at least 2 (achieved when y=1 or y=-1). Therefore, E wins, and the proposition is true.
The takeaway from this analogy is that the order in which you stack quantifiers fundamentally changes the meaning of a logical expression, and understanding this interaction can help determine the truth value of complex statements in predicate logic.
super useful, thannk you
“Math has entered the chat” :me “I was trying to see how to write a universal statement for English. “English has left the chat”
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Can i asking questions and how?
Thanks so much for this most helpful video. I've been studying maths for 16 months now. I love it. It's something I cannot not do.
Sir, can you help me with a question? Regarding quantifiers...
Let L(x, y) be the statement “x loves y,” where the domain for both x and y consists of all people in the world. Use a quantifier to express this statement.
statement: Joy does not love someone
my answer: Ey ㄱL(Joy, y)
lecturer's answer: Ex ㄱL(Joy, x)
I dont understand her answer 😅 did she sub someone labelled as x into the y part? where did I go wrong
@@DrTrefor Thank you sir! I appreciate it a lot, my favorite Math Professor 🙆♀️
Awesome
It confuse me more how he can write like that than the lecture itself
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haha, glad it helped!
@@DrTrefor so such that can be written as, too
really great
thank you
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Your explanation in easy to understand than my teacher
Glad it helped!
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It's like Plato's forms and essences but put into a symbolic notation? Well, this is cool
how do you even write like that xD btw nice lesson this refreshes me for the upcoming exam thanks
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