W5 - predicate logic

Question 1

Predicates

Answer 1

A statement with a variable where the predicate is either True or False

A predicate is a kind of function whose codomain is the set {True,False}

Question 2

How to write notation for predicate logic

Answer 2

example
∃X∀Y : f(x)

note that order matters for ∀Y and ∃X

Question 3

What is
∀X: (p(X)∧q(X))
and
∃X: (p(X)∨q(X))

Answer 3

Universal quantifiers and conjunction mix in a natural way (since we need all ‘and’)
∀X: (p(X)∧q(X)) is logically equivalent to (∀X p(X))∧(∀X q(X))

Similarly, existential quantifiers and disjunction mix naturally too. (since as need all ‘or’)
∃X: (p(X)∨q(X)) is logically equivalent to (∃X p(X))∨(∃X q(X))

Question 4

What does ¬ ∀Y : f(y) simplify to

Answer 4

¬ ∀Y: f(y) same as ∃ Y : ¬f(y)