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Assignment #2
Section 1.2 on page 19.
5b)t
e) F
f) T
7a) p => q,
(~p) v q
p  q  ~p  p => q  (~p) v q

T
T
F
T
T
F
T
T
T
T
T
F
F
F
F
F
F
T
T
T
7b) P<=>Q and (P=>Q) ^ (Q=>P)
p  q  p=>q  q=>p  p<=>q  (p=>q) ^ (q=>p)

T  T 
T

T

T

T
F  T 
T

F

F

F
T  F 
F

T

F

F
F  F 
T

T

T

T
7d) ~(P^Q) and (~P) V (~Q)
P  Q  ~P  ~Q  P^Q  ~(P ^ Q)  (~P) v (~Q)

T  T  F
 F
 T

F

F
F  T  T
 F
 T

F

F
T  F  F
 T
 T

F

F
F  F  T
 T
 F

T

T
8g)
B is invertible is a necessary and sufficient condition for det(B) <> 0.
Logical connectives:
(B is invertible) <=> (det(B) <> 0)
12a)
converse:
(f'(x) = 0) => ((f has a rel. min at x) ^ ( f is diff. at x))
False
contrapositive:
(f'(x) <>0) => ((f does have a rel. min at x) V ( f is not diff. at x))
True.
12B) Converse: If n=2 or n is odd, then n is prime.
False. 9 is odd and not
prime.
Contrapositive: If n is even and not equal to 2, then n is not prime.
True
12c)
converse:
((x is real) ^ ( x is not rational)) => ( x is irrational)
True.
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View Full Documentcontrapositive:
((x is not real) V (x is rational)) => ( x is not irrational)
True.12d) converse: (x = 1) => ((x = 1) v (x = 1))
True.contrapositive:(x neq 1) => ((x neq 1) ^ (x neq 1))
True.
14a)
P  Q  P => Q  ~P  ~Q  ~P => ~Q

T
T
T
F
F
T
T
F
F
F
T
T
F
T
T
T
F
F
F
F
T
T
T
T
14b)
P=>Q and its inverse are both true if both P and Q are either
both true or false as shown in the truth table above
14c)
Conditional Sentence
: P => Q
Converse
: Q => P
Inverse
: ~P => ~Q
Contrapositive
: ~Q => ~P
Contrapositive of inverse
: ~(~Q) => ~(~P) or Q => P
Inverse of contrapositive
: ~(~Q) => ~(~P) or Q => P
Hence, both the inverse of the contrapositive and the contrapositive
of the inverse are equivalent to the converse.
13c)
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