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?
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1) ∃𝑦∀𝑥𝑄(𝑥, 𝑦)
Since for every y we can put x = 0, and then , which means this statement is true
2) ∀𝑥∃𝑦𝑄(𝑥, 𝑦)
Since, for example, for x = 2 we have , which means this statement is false