Use De Morgan's law for quantified statements and the laws of propositional logic to show the following equivalences:
(a) Vr (P(2) A-Q(x)) = 3r (-P(2) VQ(x))
(b) Vr (-P(z) → Q(2)) = 3r (-P(2)A-Q())
(c) -3x (-P(x) V (Q(2)^-R(z))) = Vz(P(2)^(-Q(x) V R()))

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step-by-step explanation:

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guess at question:

a vertical line intersects the graph of g(x,y) = 0 at two points. what conclusion follows about the function f(x) implicitly defined by g(x,f(x)) = 0?

the domain of f(x) cannot include the x coordinate of any such vertical line.

informally, f(x) is not a function, and

g(x,y)=0 cannot be solved for y uniquely.

the relation r = {(x,y) such that g(x,y) = 0} is a function if

for all (a,b) ∊ r and (c,d) ∊ r, a=c implies b=d.

step-by-step explanation:

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step-by-step explanation:

is there supposed to be a diagram or a graph for this one? ?

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