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Question 1175639: Construct a truth table for ( p ---> ~ r ) v ( ~ q ^ r )
Found 2 solutions by Solver92311, Edwin McCravy:
Answer by Solver92311(821)   (Show Source):
Answer by Edwin McCravy(20056)  (Show Source):
You can put this solution on YOUR website!
( p ---> ~ r ) v ( ~ q ^ r )
Start with this.
| p | q | r | -q | -r | p ---> ~r | ~q ^ r || ( p ---> ~r ) v ( ~q ^ r ) |
|---|---|---|----|----|-----------|------------------------------------------
| 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 ~q and the ~r column with the opposite truth values of the q and r
columns:
| p | q | r | -q | -r | p ---> ~r | ~q ^ r || ( p ---> ~r ) v ( ~q ^ r ) |
|---|---|---|----|----|-----------|------------------------------------------
| T | T | T | F | F | | || |
| T | T | F | F | T | | || |
| T | F | T | T | F | | || |
| T | F | F | T | T | | || |
| F | T | T | F | F | | || |
| F | T | F | F | T | | || |
| F | F | T | T | F | | || |
| F | F | F | T | T | | || |
To fill in the next column, p ---> ~r, using the rule for --->.
Put T except in case of "T ---> F", [T on the left and F on the right
of --->) and put F in that case. (Notice that only the 1st and 3rd rows have T
under p and F under ~r. So put F's in those two rows and T's everywhere else:
| p | q | r | -q | -r | p ---> ~r | ~q ^ r || ( p ---> ~r ) v ( ~q ^ r ) |
|---|---|---|----|----|-----------|------------------------------------------
| T | T | T | F | F | F | || |
| T | T | F | F | T | T | || |
| T | F | T | T | F | F | || |
| T | F | F | T | T | T | || |
| F | T | T | F | F | T | || |
| F | T | F | F | T | T | || |
| F | F | T | T | F | T | || |
| F | F | F | T | T | T | || |
To fill in the next column, ~q^r, using the rule for ^.
Put F except in case of "T ---> T", [T on the both sides of ^] and put T in
that case. (Notice that only the 3rd and 7th rows have T under both ~q and r.
So put T's in those two rows and F's everywhere else:
| p | q | r | ~q | ~r | p ---> ~r | ~q ^ r || ( p ---> ~r ) v ( ~q ^ r ) |
|---|---|---|----|----|-----------|------------------------------------------
| T | T | T | F | F | F | F || |
| T | T | F | F | T | T | F || |
| T | F | T | T | F | F | T || |
| T | F | F | T | T | T | F || |
| F | T | T | F | F | T | F || |
| F | T | F | F | T | T | F || |
| F | F | T | T | F | T | T || |
| F | F | F | T | T | T | F || |
To fill in the last column, (p--->~r)v(~q^r), using the rule for v.
Put T except in case of "F ---> F", [F on the both sides of v] and put F in
that case. (Notice that only the 1st row has F under both p--->~r and ~qvr. So
put F's on the first row and T's everywhere else:
| p | q | r | ~q | ~r | p ---> ~r | ~q ^ r || ( p ---> ~r ) v ( ~q ^ r ) |
|---|---|---|----|----|-----------|------------------------------------------
| T | T | T | F | F | F | F || F |
| T | T | F | F | T | T | F || T |
| T | F | T | T | F | F | T || T |
| T | F | F | T | T | T | F || T |
| F | T | T | F | F | T | F || T |
| F | T | F | F | T | T | F || T |
| F | F | T | T | F | T | T || T |
| F | F | F | T | T | T | F || T |
Edwin
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