"As an
example, if we let x and y be variables with the domain R, and let P(x,y) be the
open sentence
P(x,y) : x>y,
then the sentence ∀x ∈ R,P(x,y) is not a statement, because it still has a variable that has not been specified or quantified (namely y). In order to make it a statement, we need to evaluate y; we could say ∀y ∈ R, ∀x ∈ R, P (x, y), which means that
For all real numbers y and for all real numbers x, it holds that x > y."
Even though the author stated that "x and y [are] variables with the domain R," because this is not explicitly stated in the claim itself ("∀x ∈ R,P(x,y)"), it is not a statement. I am assuming that the aforementioned thing in parenthesis is the only thing I am supposed to be looking at when evaluating the statement, rather than look at the paragraph as a whole. Is that correct?
P(x,y) : x>y,
then the sentence ∀x ∈ R,P(x,y) is not a statement, because it still has a variable that has not been specified or quantified (namely y). In order to make it a statement, we need to evaluate y; we could say ∀y ∈ R, ∀x ∈ R, P (x, y), which means that
For all real numbers y and for all real numbers x, it holds that x > y."
Even though the author stated that "x and y [are] variables with the domain R," because this is not explicitly stated in the claim itself ("∀x ∈ R,P(x,y)"), it is not a statement. I am assuming that the aforementioned thing in parenthesis is the only thing I am supposed to be looking at when evaluating the statement, rather than look at the paragraph as a whole. Is that correct?
Also, I was taught not to start sentences with math symbols when writing formal math things. However the book does so: "P implies Q. P only if Q. Q if P." What is the proper standard?
I am wondering what practical application that this section has. I'd imagine that it is more of a thought exercise than an actual useful tool in most cases. However, I have read formal and relevant proofs that did include statements similar to the ones about which we are learning. However, I have not seen them in nearly as much depth as the book is providing.
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