Answer to exercise #1d


Step 1

P
Q
R
~
(
~
(
P
(
~
P
Q
)
)
~
R
)
T T T
T
T
T
T
T T F
T
T
T
F
T F T
T
T
F
T
T F F
T
T
F
F
F T T
F
F
T
T
F T F
F
F
T
F
F F T
F
F
F
T
F F F
F
F
F
F


Step 2

P
Q
R
~
(
~
(
P
(
~
P
Q
)
)
~
R
)
T T T
T
F
T
T
F
T
T T F
T
F
T
T
T
F
T F T
T
F
T
F
F
T
T F F
T
F
T
F
T
F
F T T
F
T
F
T
F
T
F T F
F
T
F
T
T
F
F F T
F
T
F
F
F
T
F F F
F
T
F
F
T
F


Step 3

P
Q
R
~
(
~
(
P
(
~
P
Q
)
)
~
R
)
T T T
T
F
T
T
T
F
T
T T F
T
F
T
T
T
T
F
T F T
T
F
T
F
F
F
T
T F F
T
F
T
F
F
T
F
F T T
F
T
F
T
T
F
T
F T F
F
T
F
T
T
T
F
F F T
F
T
F
T
F
F
T
F F F
F
T
F
T
F
T
F


Step 4

P
Q
R
~
(
~
(
P
(
~
P
Q
)
)
~
R
)
T T T
T
T
F
T
T
T
F
T
T T F
T
T
F
T
T
T
T
F
T F T
T
F
F
T
F
F
F
T
T F F
T
F
F
T
F
F
T
F
F T T
F
F
T
F
T
T
F
T
F T F
F
F
T
F
T
T
T
F
F F T
F
F
T
F
T
F
F
T
F F F
F
F
T
F
T
F
T
F


Step 5

P
Q
R
~
(
~
(
P
(
~
P
Q
)
)
~
R
)
T T T
F
T
T
F
T
T
T
F
T
T T F
F
T
T
F
T
T
T
T
F
T F T
T
T
F
F
T
F
F
F
T
T F F
T
T
F
F
T
F
F
T
F
F T T
T
F
F
T
F
T
T
F
T
F T F
T
F
F
T
F
T
T
T
F
F F T
T
F
F
T
F
T
F
F
T
F F F
T
F
F
T
F
T
F
T
F


Step 6

P
Q
R
~
(
~
(
P
(
~
P
Q
)
)
~
R
)
T T T
F
T
T
F
T
T
T
T
F
T
T T F
F
T
T
F
T
T
T
T
T
F
T F T
T
T
F
F
T
F
F
F
F
T
T F F
T
T
F
F
T
F
F
T
T
F
F T T
T
F
F
T
F
T
T
F
F
T
F T F
T
F
F
T
F
T
T
T
T
F
F F T
T
F
F
T
F
T
F
F
F
T
F F F
T
F
F
T
F
T
F
T
T
F


Step 7

P
Q
R
~
(
~
(
P
(
~
P
Q
)
)
~
R
)
T T T
F
F
T
T
F
T
T
T
T
F
T
T T F
F
F
T
T
F
T
T
T
T
T
F
T F T
T
T
T
F
F
T
F
F
F
F
T
T F F
F
T
T
F
F
T
F
F
T
T
F
F T T
T
T
F
F
T
F
T
T
F
F
T
F T F
F
T
F
F
T
F
T
T
T
T
F
F F T
T
T
F
F
T
F
T
F
F
F
T
F F F
F
T
F
F
T
F
T
F
T
T
F

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