SOLUTION: Determine the truth value for the given statement when p is false, q is true, and r is false.
(~p∧q)↔~r
Algebra ->
Conjunction
-> SOLUTION: Determine the truth value for the given statement when p is false, q is true, and r is false.
(~p∧q)↔~r
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Since p = false, this makes ~p = true
Apply the conjunction ~p ^ q to get true ^ true = true
The ~p ^ q portion is true.
So is ~r because r = false and ~r = true
Both (~p ^ q) and ~r produce the same truth values of "true"
Overall, the statement (~p ^ q) <--> ~r is true when p = false, q = true, r = false
If you wanted, you could make a full truth table like so.
The row to highlight or focus on would be row 6 marked in red.
p
q
r
~p
~r
~p ^ q
(~p ^ q) <--> ~r
T
T
T
F
F
F
T
T
T
F
F
T
F
F
T
F
T
F
F
F
T
T
F
F
F
T
F
F
F
T
T
T
F
T
F
F
T
F
T
T
T
T
F
F
T
T
F
F
T
F
F
F
T
T
F
F
and this is what the condensed table looks like when focusing on row 6 only
p
q
r
~p
~r
~p ^ q
(~p ^ q) <--> ~r
F
T
F
T
T
T
T
I used spreadsheet software to generate the truth tables quickly.
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