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2. Conditional. Components: One is lucky, a solitary fantasy can totally transform one million realities; →
3. Disjunction. Components: He didn’t get my message, he’s ignoring me; v
4. NO conjunction, disjunction, or conditional; simply negation.
Exercise 7.2
2. y → x
3. p v q
6. ~r
7. p & ~q
10. ~y & ~z
15. ~a & b
Exercise 7.3
3. False
5. True
7. True
10. True
Exercise 7.4
1. True
3. False
7. True
10. True
Exercise 7.5
1. If alligators are reptiles, then alligators have scales.
7. Water is wet, and fire is not cold.
8. Either the bakery is not open, or the pub is not open.
10. If the day goes well, then we will not regret our efforts.
Exercise 7.6
3. p v q
4. p → q
7. ~p & ~q
10. p & ~q
Exercise 7.7
1. p → q
p |
q |
p → q |
T |
T |
T |
T |
F |
F |
F |
T |
T |
F |
F |
T |
5. p v q
p |
q |
p v q |
T |
T |
T |
T |
F |
T |
F |
T |
T |
F |
F |
F |
7. ~p → q
p |
q |
~p |
~p → q |
T |
T |
F |
T |
T |
F |
F |
T |
F |
T |
T |
T |
F |
F |
T |
F |
10. ~~ p v q
p |
q |
~~p |
~~p v q |
T |
T |
T |
T |
T |
F |
T |
T |
F |
T |
F |
T |
F |
F |
F |
F |
Exercise 7.8
2. Invalid
p v q
q
∴ ~ p
p |
q |
p v q |
q |
~p |
T |
T |
T |
T |
F |
T |
F |
T |
F |
F |
F |
T |
T |
T |
T |
F |
F |
F |
F |
T |
5. Invalid
p → q
~ p
∴ q
p |
q |
p → q |
~p |
q |
T |
T |
T |
F |
T |
T |
F |
F |
F |
F |
F |
T |
T |
T |
T |
F |
F |
T |
T |
F |
6. Invalid
q → r
p → q
∴ q
p |
q |
r |
q → r |
p → q |
q |
T |
T |
T |
T |
T |
T |
T |
T |
F |
F |
T |
T |
T |
F |
T |
T |
F |
F |
T |
F |
F |
T |
F |
F |
F |
T |
T |
T |
T |
T |
F |
T |
F |
F |
T |
T |
F |
F |
T |
T |
T |
F |
F |
F |
F |
T |
T |
F |
8. Valid
a v (b & c)
~ (b & c)
∴ a
a |
b |
c |
(b & c) |
a v(b&c) |
~ (b & c) |
a |
T |
T |
T |
T |
T |
F |
T |
T |
T |
F |
F |
T |
T |
T |
T |
F |
T |
F |
T |
T |
T |
T |
F |
F |
F |
T |
T |
T |
F |
T |
T |
T |
T |
F |
F |
F |
T |
F |
F |
F |
T |
F |
F |
F |
T |
F |
F |
T |
F |
F |
F |
F |
F |
F |
T |
F |
9. Valid
p → q
∴ p → (p & q)
p |
q |
p & q |
p → q |
p → (p&q) |
T |
T |
T |
T |
T |
T |
F |
F |
F |
F |
F |
T |
F |
T |
T |
F |
F |
F |
T |
T |
10. Valid
x → y
y → z
∴ x → z
x |
y |
z |
x → y |
y → z |
x → z |
T |
T |
T |
T |
T |
T |
T |
T |
F |
T |
F |
F |
T |
F |
T |
F |
T |
T |
T |
F |
F |
F |
T |
F |
F |
T |
T |
T |
T |
T |
F |
T |
F |
T |
F |
T |
F |
F |
T |
T |
T |
T |
F |
F |
F |
T |
T |
T |
13. Valid
(p v q) → (p & q)
p & q
∴ p v q
p |
q |
(p v q) |
(p & q) |
(p v q) →(p & q) |
p & q |
p v q |
T |
T |
T |
T |
T |
T |
T |
T |
F |
T |
F |
F |
F |
T |
F |
T |
T |
F |
F |
F |
T |
F |
F |
F |
F |
T |
F |
F |
15. Valid
(d v e) → (d & e)
~ (d v e)
∴ ~ (d & e)
d |
e |
(d v e) |
(d & e) |
(d v e) → (d & e) |
~ (d v e) |
~ (d & e) |
T |
T |
T |
T |
T |
F |
F |
T |
F |
T |
F |
F |
F |
T |
F |
T |
T |
F |
F |
F |
T |
F |
F |
F |
F |
T |
T |
T |
16. Valid
(p → q) → (p → r)
~ (p → q)
~ r
∴ p
p |
q |
r |
(p → q) |
(p → r) |
(p →q) → (p → r) |
~ (p →q) |
~r |
p |
T |
T |
T |
T |
T |
T |
F |
F |
T |
T |
T |
F |
T |
F |
F |
F |
T |
T |
T |
F |
T |
F |
T |
T |
T |
F |
T |
T |
F |
F |
F |
F |
T |
T |
T |
T |
F |
T |
T |
T |
T |
T |
F |
F |
F |
F |
T |
F |
T |
T |
T |
F |
T |
F |
F |
F |
T |
T |
T |
T |
F |
F |
F |
F |
F |
F |
T |
T |
T |
F |
T |
F |
Exercise 7.9
1. Valid
p → q
~ p → r
~ p
∴ r
p |
q |
r |
~ p → r |
~ p |
r |
T |
T |
T |
T |
F |
T |
T |
T |
F |
T |
F |
F |
T |
F |
T |
T |
F |
T |
T |
F |
F |
T |
F |
F |
F |
T |
T |
T |
T |
T |
F |
T |
F |
F |
T |
F |
F |
F |
T |
T |
T |
T |
F |
F |
F |
F |
T |
F |
2. Valid
p → q
~q
∴ ~ p
p |
q |
p → q |
~q |
~p |
T |
T |
T |
F |
F |
T |
F |
F |
T |
F |
F |
T |
T |
F |
T |
F |
F |
T |
T |
T |
6. Invalid
f → ~ s
~s → p
∴ p
f |
s |
p |
f → ~s |
~s → p |
p |
T |
T |
T |
F |
T |
T |
T |
T |
F |
F |
T |
F |
T |
F |
T |
T |
T |
T |
T |
F |
F |
T |
F |
F |
F |
T |
T |
T |
T |
T |
F |
T |
F |
T |
T |
F |
F |
F |
T |
T |
T |
T |
F |
F |
F |
T |
F |
F |
7. Valid
p → q
p
∴ q
p |
q |
p → q |
p |
q |
T |
T |
T |
T |
T |
T |
F |
F |
T |
F |
F |
T |
T |
F |
T |
F |
F |
T |
F |
F |
10. Valid
w v ~ w
~ w → ~ r
w → r
~ w
∴ ~ r
w |
r |
w v ~ w |
~ w → ~ r |
w → r |
~ w |
~ r |
T |
T |
T |
T |
T |
F |
F |
T |
F |
T |
T |
F |
F |
T |
F |
T |
T |
F |
T |
T |
F |
F |
F |
T |
T |
T |
T |
T |
Exercise 7.10
5. Invalid
p → q
~ p
∴ q
T | T | F |
p → q | ~ p | q |
F F | F | F |
10. Valid
x → y
y → z
∴ x → z
F | T | F |
x → y | y → z | x → z |
T F | F F | T F |
18. Invalid
~ (d & e)
e v f
∴ ~ d & f
T | T | F |
~ (d & e) | e v f | ~ d & f |
T F | F T | T T |