Practice problem on Translations

Question

Given a statement: "At least 1 pigeon-hole has at least 2 pigeons in it".

and functions:

ph(x) : x is pigeon-hole

p(x) : x is pigeon

h(x,y) : x has y

Answer the following questions:

1. Translate the given statement into logical expression.
2. What would be negation of the logical expression we get in part (a).
3. Translate logical expression we get in part (b) to english.

Answer 1

a) ∃x(ph(x) ∧ ∃y(p(y) ∧ h(x,y) ∧ ∃z(p(z) ∧ h(x,z))))

b) ∀x(ph(x) → ¬(∃y(p(y) ∧ h(x,y) ∧ ∃z(p(z) ∧ h(x,z)))))

c) "For all pigeon-holes, if it exists, it does not have at least two pigeons in it."