I am starting to learn math proofs.
Is there a difference between writing "s.t." and writing the commas ","?
For example, one of my homework questions is:
Let S = [1, 2] and T = (3, ∞). Determine the truth value of the following statements. Explain your answers carefully.
(a) ∃x ∈ S s.t. ∃y ∈ T s.t. |x − y| > 3
(b) ∀x ∈ S, ∀y ∈ T , |x − y| > 3
For (a), I'm not sure why it says "∃x ∈ S s.t. ∃y ∈ T" because it seems that they are relating two different variables (x and y). How can it be that "There exists an x in S such that there exists a y in T"?
Here is what I got to answer the questions above:
(a) True because for x=1 and y=10, |x − y| > 3.
(b) False, because for x=2 and y=4, |x − y| <= 3.
Thanks in advance for any help,
A comma is a means to separate two thoughts, it doesn't mean anything more than that. The words such that (or s.t.) means "such that". This is a matter of English not math. With quantifier logic if I list two quantified statements one after the other there is an implied "such that" between all of them, so there's no need to write it, but that has to do with the fact that there are nested quantifiers involved, NOT a general thing. IN FACT, you don't even need to put a comma between quantifiers at all:
is read "There exists x such that there exists y such that P is true" or "There exists x and y such that P is true".
One needn't write a comma at all.
Further: do you realize why your solutions do indeed prove that the statements have the truth values you say they do?