Select the Best Translation into Predicate Logic, Using the Following $\lor$

question 227

Multiple Choice

select the best translation into predicate logic, using the following translation key:
b: Bhavin
c: Chloe
m: Megha
n: Nietzsche
p: Plato
Ax: x is an altruist
Jx: x is joyful
Px: x is a philosopher
Rx: x is Russian
Tx: x is thoughtful
Bxy: x is a brother of y
Mxy: x mocks y
Rxy: x is richer than y
Sxy: x is smarter than y
-Exactly three philosophers mock Plato and Nietzsche.

• A) (∃x)(∃y)(∃z){Px • Mxp • Mxn • Py • Myp • Myn • Pz • Mzp • Mzn • x≠y • x≠z • y≠z •
• (∀w)[(Pw • Mwp • Mwn) ⊃ (w≠x \( \lor \) w≠y \( \lor \) w≠z)]}
• B) (∃x)(∃y)(∃z){Px • Mxp • Mxn • Py • Myp • Myn • Pz • Mzp • Mzn • x=y • x=z • y=z •
• (∀w)[(Pw • Mwp • Mwn) ⊃ (w=x \( \lor \) w=y \( \lor \) w=z)]}
• C) (∃x)(∃y)(∃z){Px • Mxp • Mxn • Py • Myp • Myn • Pz • Mzp • Mzn • x=y • x=z • y=z •
• (∃w)[(Pw • Mwp • Mwn) • (w=x \( \lor \) w=y \( \lor \) w=z)]}
• D) (∃x)(∃y)(∃z){Px • Mxp • Mxn • Py • Myp • Myn • Pz • Mzp • Mzn • x≠y • x≠z • y≠z •
• (∀w)[(Pw • Mwp • Mwn) ⊃ (w=x \( \lor \) w=y \( \lor \) w=z)]}
• E) (∃x)(∃y)(∃z){Px • Mxp • Mxn • Py • Myp • Myn • Pz • Mzp • Mzn • x≠y • x≠z • y≠z •
• (∃w)[(Pw • Mwp • Mwn) • (w≠x \( \lor \) w≠y \( \lor \) w≠z)]}