asigurato7

2022-07-16

Writing statements into symbols Discrete Math.
The variable x represents students, F(x) means "x is a freshman", and M(x) means "x is a math major"
a) some freshme are math majors? $\mathrm{\exists }x:F\left(x\right)\phantom{\rule{thickmathspace}{0ex}}⟹\phantom{\rule{thickmathspace}{0ex}}M\left(x\right)$
b) Every math major is a freshman? $\mathrm{\forall }x:M\left(x\right)\phantom{\rule{thickmathspace}{0ex}}⟹\phantom{\rule{thickmathspace}{0ex}}M\left(x\right)$
c) No math major is a freshman? $\mathrm{¬}\mathrm{\forall }x:M\left(x\right)\phantom{\rule{thickmathspace}{0ex}}⟹\phantom{\rule{thickmathspace}{0ex}}F\left(x\right)$

Kitamiliseakekw

Expert

Step 1
(a) Some freshmen are math majors ∼ There exist x such that (x is a Freshman and x is a math major): $\mathrm{\exists }F\left(F\left(x\right)\wedge M\left(x\right)\right)$
Step 2
(b) Every math major is a freshman. ∼ For all x (if x is a math major, then x is a Freshman.) $\mathrm{\forall }x\left(M\left(x\right)\to F\left(x\right)\right)$
Step 3
(c): No math major is a freshman. ∼ There does not exist an x such that (x is a math major and x is a freshman).
$\begin{array}{rl}\mathrm{¬}\mathrm{\exists }x\left(M\left(x\right)\wedge F\left(x\right)\right)& \equiv \mathrm{\forall }x\left(\mathrm{¬}M\left(x\right)\vee \mathrm{¬}F\left(x\right)\right)\\ \\ & \equiv \mathrm{\forall }x\left(M\left(x\right)\to \mathrm{¬}F\left(x\right)\right)\end{array}$

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