Discrete Mathematics
1.3.54
(a)
p p↓p ¬p
T F F
F T T
Therefore, p↓p is logically equivalent to ¬p
(b)
p q p↓q (p↓q)↓(p↓q
)
p∨q
T T F T T
T F F T T
F T F T T
F F T F F
Therefore,(p↓q)↓(p↓ q) is logically equivalent to p∨q
(c)Because {¬,∨} is a functionally complete collection of logical
operators, and we have: ¬p≡p↓p and p∨q≡(p↓q)↓(p↓ q), so {↓} is a
functionally complete collection of logical operators.
1.4.24
(c)(i)∃x¬S(x),S(x):x can swim
(ii)∃x(C(x)∧¬S(x)),C(x):x is a person in your class
(d)(i)∀xQ(x),Q(x):x can solve quadratic equations
(ii)∀x(C(x)→Q(x))
(e)(i)∃x¬R(x),R(x):x wants to be rich
(ii)∃x(C(x)∧¬R(x))
