Question

Let 𝑄(𝑥, 𝑦) denote "𝑥 + 𝑦 = 𝑦“. What are the truth values of the quantifications ∃𝑦∀𝑥𝑄(𝑥, 𝑦) and ∀𝑥∃𝑦𝑄(𝑥, 𝑦) where the domain for all variables consists of all real numbers?

Answer 1

1) ∃𝑦∀𝑥𝑄(𝑥, 𝑦)

Since for every y we can put x = 0, and then $0 + y = y\implies y=y$ , which means this statement is true

2) ∀𝑥∃𝑦𝑄(𝑥, 𝑦)

Since, for example, for x = 2 we have $2 + y = y\implies 0=2$ , which means this statement is false