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()))

Do graduates from uf tend to have a higher income than students at fsu, five years after graduation? a random sample of 100 graduates was taken from both schools. let muf be the population mean salary at uf and let mufsu be the population mean salary at fsu. how should we write the alternative hypothesis?

Abby is preparing fruit punch for her party. she starts with a bowl filled with liquid that has 1/2 fruit juice and 1/2 sprite, but she decides it's too fruity. so she pours out 1/4 of the liquid, and then fills the bowl up again with sprite. what fraction of the punch is now sprite?

Which is a true statement comparing the graphs of x^2/6^2-y^2/8^2 = 1 and x^2/8^2-y^2/6^2 the foci of both graphs are the same points. the lengths of both transverse axes are the same. the directrices of = 1 are horizontal while the directrices of = 1 are vertical. the vertices of = 1 are on the y-axis while the vertices of = 1 are on the x-axis.