MHB Solve Logic Puzzle: Predicates & True/False Explained

[Plonker1]

View attachment 6460

I'm unsure about these three, here are my attempts. Please also explain the difference between a predicate and true/false. I assumed it is a predicate when it can be either true or false.

a) Predicate. Negation is ¬(∃n ∈ N n²>n)
b) True. Negation is, "When x<0 there is y such that y^2=x
c) No clue :P

Your help is truly appreciated!

 

[johng1]

I assume by predicate, it is meant that the sentence has unbound variables. So

a) This is a statement. The negation is
$$\exists n \text{ such that }n\in N \text{ and }n^2\leq n$$
The negation is true (set n=1) and so the original statement is false.

b) The variable x is not bound, so this is a predicate. To make a statement one might say:
$$\forall x \in R\text{ if }x<0 \text{ then }\not\exists y\in R \text{ such that }y^2=x$$
I leave it to you to negate this statement.

c) This is a true statement. ("famous" theorem of Fermat says a prime is the sum of two squares iff the prime is congruent to 1 mod 4)
The negation is:
$$\exists p\in P \text{ such that }\exists n\in N \text{ with }p=4n+1\text{ and }\forall a\in N\,\forall b\in N\,a^2+b^2\neq p$$