Hi

I am trying to answer the question which I have attached. Please tell me if the answers are correct.

a) let M(x,y) = x has forgiven y R(x)= x is a saint

U=universe of discourse= all humans

$\displaystyle \forall x \left[\exists y M(x,y)\Rightarrow R(x) \right]$

b)let R(x) = x is in calculus class ; Q(x) = x is in discrete math class

M(x,y)= x is smarter than y

U=universe of discourse = all humans

$\displaystyle \forall x \left[R(x)\Rightarrow \neg\left(\forall y\left[Q(y)\RightarrowM(x,y)\right]\right)\right]$

c)let P(x) = x likes mary ; R(x) = x is mary

U=universe= all humans

$\displaystyle \left[\forall x \left(\neg R(x)\Rightarrow P(x)\right)\right]\wedge\left[\forall x\left( R(x)\Rightarrow \neg P(x)\right)\right]$

d)let P(x) = x is a police officer

Q(x) = x is jane

R(x)= x is roger

M(x,y) = x saw y

U=universe= all humans

$\displaystyle \forall x \left[ Q(x)\Rightarrow \exists y\left ( P(y)\wedge M(x,y) \right)\right]\wedge \forall x \left[ R(x)\Rightarrow \exists y \left( P(y) \wedge M(x,y) \right ) \right]$

e)let P(x) = x is police officer

M(x,y) = x saw y

let j=jane and r=roger

again U=universe of discourse = all humans

$\displaystyle \exists y \left( P(y)\wedge M(j,y)\wedge M(r,y) \right)$

thanks