The commas are just like in normal English, they are there as a pause to separate different clauses.
For your first two check your last bit again, do they say "x is greater than y"?
For the second one you are right, it is false, as this would say that there is ONE integer which is SMALLER than ALL rationals, but the integers are rational, so this would say that there is a SMALLEST INTEGER, which is clearly false, you can always find a smaller integer than n by taking n-1.
Again, try reading the third one out, it says every odd integer has the property that four divides the difference of the integer and its cube.
Thanks for your explanation. You are right. I misread the last part for the first two where the x is smaller than y
So just to confirm, the first one is true since the statement is possible. However, the second one is false since there is no "smallest" integer as what the statement says. We can always deduct 1 unit from it and get a smaller one.
As for the third one, i am still not clear about the truth of the statement, but i think this statement is false since not all odd numbers would fit in that property. for example if x is 5, then 4|(5-5^3)= -0.0333
does that make any sense?
For negation of the statements, if i am not mistaken ∀ becomes ∃ but would the comma or any other thing change?
Negation of #1 is:
1) ∃x ∈ ℤ, ∀y ∈ ℚ, x < y.
Negation of #2 is:
2) ∀x ∈ ℤ, ∃y ∈ ℚ, x < y.
I think that does not make any sense. does it?