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)
Best Answer
$a) True$
$b) False$
$c) True$
$d) True$
$e) False$
$f) False$
$g) False$
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.