Let P(x, y) be the predicate “y = 2x”. Consider the statements...

[nikosan]

Let P(x, y) be the predicate “y = 2x”. Consider the statements...

can someone please double check that i'm on the right track with these questions?

Let P(x, y) be the predicate “y = 2x”. Consider the statements
(a) ∀x∃yP(x, y)
(b) ∀y∃xP(x, y)
(c) ∃y∀xP(x, y)

where x and y range over the integers.Write whether each statement is true or false and give a very short explanation of why

so for (a) I wrote True, because all integers can be multiplied by two to equal another integer.

(b) I wrote False, because if y = 1 then there is no integer x where y can equal 2x

(c) I wrote False, because there is no integer y for which all x makes y = 2x true.

 

[pka]

can someone please double check that i'm on the right track with these questions?
Let P(x, y) be the predicate “y = 2x”. Consider the statements
(a) ∀x∃yP(x, y)
(b) ∀y∃xP(x, y)
(c) ∃y∀xP(x, y)
where x and y range over the integers.Write whether each statement is true or false and give a very short explanation of why so for
(a) I wrote True, because all integers can be multiplied by two to equal another integer.

(b) I wrote False, because if y = 1 then there is no integer x where y can equal 2x

(c) I wrote False, because there is no integer y for which all x makes y = 2x true.

Your answers are correct but are very literal translations. We encourage students to use a natural way of speaking.
a) Two times any integer is an integer.
b) Every integer is two times some integer.
c) Some integer is two times every integer.