Answer
a) ¬M (Chou, Koko)
b) ¬M(Arlene, Sarah)∧ ¬T (Arlene, Sarah)
c) ¬M (Deborah, Jose)
d) ∀x M(x, Ken)
e) ∀x ¬T (x, Nina)
f) ∀x T(x, Avi)∨M(x, Avi))
g) ∃x∀y (y \ne x → M(x, y))
h) ∃x∀y (y \ne x → (M(x, y) ∨ T (x, y)))
i) ∃x∃y (x \ne y ∧ M(x, y) ∧M(y, x))
j) ∃x M(x, x)
k) ∃x∀y (x \ne y → (¬M(x, y) ∧¬T (y, x)))
l) ∀x (∃y (x \ne y ∧ (M(y, x) ∨ T (y, x))))
m) ∃x∃y (x \ne y ∧ M(x, y) ∧ T (y, x))
n) ∃x∃y (x \ne y ∧∀z((z = x ∧ z = y) → (M (x, z) ∨ M (y, z) ∨ T (x, z) ∨ T (y, z))))
Work Step by Step
Let M(x, y) be “x has sent y an e-mail message” and T (x, y) be “x has telephoned y,” where the domain consists of all students in your class.
for (b) it is as follows:
¬ ( M(Arlene, Sarah) ∨ T (Arlene, Sarah) )
Applying DeMorgans Law it becomes:
¬M(Arlene, Sarah)∧ ¬T (Arlene, Sarah)
