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Question 539617: I have to make the following compound statement into a truth table and i do not understand how to construct the table.
q ∨ (p → ∼r)
Answer by AnlytcPhil(1806)   (Show Source):
You can put this solution on YOUR website!
The parts that make up that expression q V (p -> ~r) are
p, q, r, ~r, (p -> ~r), and q V (p -> ~r). Make a heading for each of these.
When there are three variables, the truth table will have 8 rows.
Start with this:
p | q | r | ~r | (p -> ~r) | q V (p -> ~r)
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| | | | |
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Fill in the p column with the first half T's and last half F's
p | q | r | ~r | (p -> ~r) | q V (p -> ~r)
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T | | | | |
T | | | | |
T | | | | |
T | | | | |
F | | | | |
F | | | | |
F | | | | |
F | | | | |
Fill in the q column with 2 T's, 2 F's, 2 T's, 2 F's:
p | q | r | ~r | (p -> ~r) | q V (p -> ~r)
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T | T | | | |
T | T | | | |
T | F | | | |
T | F | | | |
F | T | | | |
F | T | | | |
F | F | | | |
F | F | | | |
Fill in the r-column alternating T,F,T,F,T,F,T,F
p | q | r | ~r | (p -> ~r) | q V (p -> ~r)
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T | T | T | | |
T | T | F | | |
T | F | T | | |
T | F | F | | |
F | T | T | | |
F | T | F | | |
F | F | T | | |
F | F | F | | |
Fill in the ~r column as the opposite of the r column
p | q | r | ~r | (p -> ~r) | q V (p -> ~r)
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T | T | T | F | |
T | T | F | T | |
T | F | T | F | |
T | F | F | T | |
F | T | T | F | |
F | T | F | T | |
F | F | T | F | |
F | F | F | F | |
Fill in the (p -> ~r) column by this rule: If the
p column has a T and the ~r column has a F, then put
an F in the (p -> ~r) column; otherwise put a T.
p | q | r | ~r | (p -> ~r) | q V (p -> ~r)
------------------------------------------
T | T | T | F | F |
T | T | F | T | T |
T | F | T | F | F |
T | F | F | T | T |
F | T | T | F | T |
F | T | F | T | T |
F | F | T | F | T |
F | F | F | F | T |
Fill in the q V (p -> ~r) by this rule:
If both the q column and the (p -> ~r) column
have F's, then put an F in the q V (p -> ~r)
column; otherwise put T.
p | q | r | ~r | (p -> ~r) | q V (p -> ~r)
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T | T | T | F | F | T
T | T | F | T | T | T
T | F | T | F | F | F
T | F | F | T | T | T
F | T | T | F | T | T
F | T | F | T | T | T
F | F | T | F | T | T
F | F | F | F | T | T
Edwin
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