SOLUTION: Construct a truth table for ( p ---> ~ r ) v ( ~ q ^ r )

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): You can put this solution on YOUR website!

p q ~q r ~r (p -> ~r) (~q ^ r) [(p -> ~r) V (~q ^ r)} T T F T F F F F T T F F T T F T T F T T F F T T T F T F T T F T F T F T F T F T F T F F T T F T F F T T F T T T F F T F T T F T

John

My calculator said it, I believe it, that settles it

From
I > Ø

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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