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I'm learning how to determine the truth value of statements and I want to make sure that i'm understanding and answering the questions correctly. I'm struggling with determining if i'm reading the statements correctly. I'm reading $\forall$x $\exists$y as "for all x there exists a y". Is this correct? Are my answers correct? (my answers are the italics and the problem sets are to the left)

Domain: $\mathbb R$ (all real numbers)

a) ∀x∃y(x^2 = y) = True (for any x^2 there is a y that exists)

b) ∀x∃y(x = y^2) = False (x is negative no real number can be negative^2

c) ∃x∀y(xy=0) = True (x = 0 all y will create product of 0)

d) ∀x(x≠0 → ∃y(xy=1)) = True (x != 0 makes the statement valid in the domain of all real numbers)

e) ∃x∀y(y≠0 → xy=1) = False (no single x value that satisfies equation for all y

f) ∃x∃y(x+2y=2 ∧ 2x+4y=5) = False (doubling value through doubling variable coefficients without doubling sum value)

g) ∀x∃y(x+y=2 ∧ 2x−y=1) = False (really unsure about this one)

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    Not "there is a y that exists" but "there is a y" **or** "exists a y". – 2017-02-10
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    @MauroALLEGRANZA got it. That makes more sense "exists a y". Do my answers conform with that logic? – 2017-02-10
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    @MauroALLEGRANZA and thank you for your help! – 2017-02-10
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    g) consider it as a system of equations; adding them we get : $3x=3$ that is not true for every $x$. Proof: consider $x=2$; then we must have some $y$ such that $2+y=2$ and $4-y=1$, which is impossible. Conclusion: *FALSE*. – 2017-02-10
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    Other than slight roughness in the wording (which I expect will improve as you read more mathematics), the thinking appears to be good. For g), you just need one counterexample. An easy one is $x=0.$ (In fact you only needed one counterexample for b), although you found infinitely many.) – 2017-02-10
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    Thank you gents! Just starting out with this area and am very excited to keep progressing forward. – 2017-02-10

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$a) True$

$b) False$

$c) True$

$d) True$

$e) False$

$f) False$

$g) False$

Basically you got them all right. Regarding $g)$, you can either find an $x$ for which the statement isn't true or you solve the equation, obtaining specific (static, say) values for $x$ and $y$, thus proving it's falsity.