I need to do the negation of ∃y∀x(x ≥ y +1) using generalized de morgan's law to simplify the expression. This is what I have;
¬(∃y∀x(x ≥ y +1)) ≡ ∀y¬(∀x(x≥y +1)) ≡ ∀y∃x¬(x ≥ y +1) ≡ ∀y∃x(x < y +1)
I'm pretty sure I've done it all right except I'm not sure about the last statement in particular the < sign and the +1.
Okay that's my main problem that I would love to get some help on but if someone could also tell me if the following is correct.
For ∃y∀x(x ≥ y +1) I need to prove whether the statement is true or not if the domain of discourse is Z+ which I'm pretty sure I have the right answer for (this isn't what I want clarify) but the question is I need to state a pair of domains (one for x, one for y) for which the proposition is true. Would I be correct with the following?~
5 for x, 4 for y as 5 ≥ 4 + 1 ≡ 5 ≥ 5 is true.
Thanks for the help!
Really appreciate the help, I'm quite surprised that ≡ ∀y∃x(x < y +1) is right I thought it might be something like ≡ ∀y∃x(x < y -1).
Anyway for the second part the question is worded exactly "(c) State a pair of domains (one for x, one for y) for which the proposition is true." and I have to use the domain of discourse Z+ so I can't use 0 but you're telling me that domains is in fact a pair of sets I need to state {1, 2 ...} for y and {2, 3 ...} for x. Is using "..." correct for demonstrating the pattern +1 every time?
Thanks again.
Edit: P.S. I'm a complete math noob, I didn't even do math from year 11 in high school and this is the only math course I need to do in my degree so I apologize if what I'm asking is obvious.