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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Question 1204413: Determine the truth value for the given statement when p is false, q is true, and r is false.
(~p∧q)↔~r
Answer by math_tutor2020(3817) (Show Source): You can put this solution on YOUR website!
Answer: true
Explanation
Here is a review of various truth table rules
https://www.algebra.com/algebra/homework/Conjunction/truth-table1.lesson
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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