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The variable $x$ represents stduents, $F(x)$ means "$x$ is a freshman", and $M(x)$ means "$x$ is a math major"

a) some freshme are math majors? $\exists x:F(x) \implies M(x)$

b) Every math major is a freshman? $\forall x:M(x) \implies M(x)$

c) No math major is a freshman? $\neg\forall x:M(x) \implies F(x)$

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$(a)$ Some freshmen are math majors
$\sim$ There exist x such that (x is a Freshman and x is a math major):

$$\exists F(F(x) \land M(x))$$

$(b)$ Every math major is a freshman.
$\sim$ For all x (if x is a math major, then x is a Freshman.) $$ \forall x (M(x) \rightarrow F(x))$$

$(c):$ No math major is a freshman.
$\sim$ There does not exist an x such that (x is a math major and x is a freshman). $$\begin{align} \lnot \exists x(M(x) \land F(x)) & \equiv \forall x(\lnot M(x) \lor \lnot F(x))\\ \\ &\equiv \forall x(M(x) \rightarrow \lnot F(x))\end{align}$$

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Right up your alley +1 – Amzoti Dec 7 '13 at 0:20

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