Could someone please check my answers for these questions:

p(x): x^2 - 7x + 10 = 0; x = {5, 2}

q(x): x^2 - 2x - 3; x = {-1, 3}

r(x): x < 0

domains:

i) Z (set of all integers)

ii) Z+ (set of all positive integers)

iii) integers 2 and 5

Are these statements True or false (with counterexamples):

$\displaystyle

\forall{x[p(x)\Rightarrow\neg}{r(x)]}

$

i) false x = 1

ii) false x = 1

iii) false

$\displaystyle

\forall{x[q(x)\Rightarrow}{r(x)]}

$

i) false x = 3

ii) false x = 3

iii) false x = 2 ?

$\displaystyle

\exists{x[q(x)\Rightarrow}{r(x)]}

$

i) true

ii) false

iii) false

$\displaystyle

\exists{x[p(x)\Rightarrow}{r(x)]}

$

i) false

ii) false

iii) false

Is it possible to find counterexamples for the existential statements? Also, in the first two problems with the universal quantifier, for these to be true, would p(x) and q(x) have to hold for every value in the domain? For example, would x^2 - 7x + 10 = 0 have to be true for every value of x in the domain, or just the solutions, 5 and 2? I understand quantifiers, but the conditional statement is confusing me a bit. Thanks!