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Use De Morgans law for quantied statements and the laws of propositional logic to show the following equivalences:

please do not do it on paper

here is an example on how it should be formated:

x(P(x)Q(x))andx(P(x) Q(x))

x(P(x)Q(x)))De Morgans law

x(P(x)Q(x))Conditional identity

x(P(x)Q(x))Double negation law

x(P(x) Q(x))De Morgans law

a)

(P(x) Q(x)) x(P(x) Q(x))

b)

x(P (x) = Q(x)) x(P (x) Q(x))

c)

(P (x) v (Q(x) R(x))) x(P (x) (Q(x) R(x)))