Question

For integers 𝑥 and 𝑦, consider the proposition ∃𝑥∀𝑦 (𝑥 + 𝑦 = 10). Is the proposition true, false, or does it depend on the values of 𝑥 and 𝑦?

Answer 1

The proposition $∃𝑥∀𝑦 (𝑥 + 𝑦 = 10)$ is false because of for any integer number $x$ there exists the integer number $y=-x$ such that $x+y=x-x=0\ne 10.$