Universal and Existential Quantifiers, ∀ 'For All' and ∃ 'There Exists'

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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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I am grateful for the clarity of your breakdown of the problem. Thank you

adamloepker

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tianhaowang

Your explantions and examples about the universal and existential quantifiers are so well.Thanks.

jeandedieunzambi

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.

KnakuanaRka

Thank you for making these videos. Your explanation is a lot easier to understand!

dta_yoon

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!!!

Salvation