# Translate the following statement into mathematical English.

Complete the following problems:
a) Translate the following statement into mathematical English. $\mathrm{\forall }x\left[\left(P\left(x\right)\wedge ⌝\left(x=2\right)\right)\to O\left(x\right)\right]$, where P(x) means "r is a prime number" and O(x) means "x is odd"
b) Negate this statemem and express the result as a positive satemem (without using "no" or "nor"): There issomeone in the freshman class who doesn't have a roommate
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Step 1
For part (a) $\mathrm{\forall }x$ (If xis a prime number and x is not equal to 2 then x is an odd number)
Every prime number except 2 is an odd number.
Step 2
For part (b)
$F\left(x\right)=x$ is in the freshman class
$R\left(x,y\right)=$ x has roommate y
Original Statement is $\mathrm{\exists }xF\left(x\right)⌝\mathrm{\exists }yR\left(x,y\right)$
Negation will be as follows:
$-\mathrm{\exists }xF\left(x\right)\wedge \mathrm{\exists }yR\left(x,y\right)$
$\mathrm{\forall }x⌝\left(F\left(x\right)\wedge \mathrm{\exists }yR\left(x,y\right)$
$\mathrm{\forall }x⌝F\left(x\right)\vee \mathrm{\exists }yR\left(x,y\right)$
$\mathrm{\forall }x⌝F\left(x\right)\vee \mathrm{\exists }yR\left(x,y\right)$
$\mathrm{\forall }xF\left(x\right)\to \mathrm{\exists }yR\left(x,y\right)$
Everyone in the freshman class has atleast one roommate