Question 1083625: P ∙ (Q ⊃ R) ; (P ∙ Q) ⊃ (P ∙ R)
are logically equivalent to each other, or whether they are contradictory to each other by making a truth table for them. If they are neither of those, determine whether they are consistent with each other, or whether they are inconsistent with each other
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website! T = True
F = False
I'm using the ampersand in place of the dot (to make it more visible).
Also I'm using the symbol ">" without quotes in place of the horsehoe.
Truth Table for P & (Q > R)
| P | Q | R | Q > R | P & (Q > R) |
|---|
| T | T | T | T | T | | T | T | F | F | F | | T | F | T | T | T | | T | F | F | T | T | | F | T | T | T | F | | F | T | F | F | F | | F | F | T | T | F | | F | F | F | T | F |
Truth Table for (P & Q) > (Q & R)
| P | Q | R | P & Q | P & R | (P & Q) > (P & R) |
|---|
| T | T | T | T | T | T | | T | T | F | T | F | F | | T | F | T | F | T | T | | T | F | F | F | F | T | | F | T | T | F | F | T | | F | T | F | F | F | T | | F | F | T | F | F | T | | F | F | F | F | F | T |
Carefully compare (row by row) the last columns of each table. Here's a side by side comparison of the last two columns of each table.
| P & (Q > R) | (P & Q) > (P & R) |
|---|
| T | T | | F | T | | T | T | | T | T | | T | T | | F | T | | T | T | | T | T |
as you can see the columns do not match up. Therefore, the two logical expressions aren't equivalent.
They aren't contradictory to one another because sometimes they do match up (like in row 1 we have two T's) but other times they don't (row 2 with F and then T). For a contradiction, the values need to be opposite one another (one must be true and the other false or vice versa) and this needs to apply to every row.
The two expressions are consistent because it is possible for them both to be true under the same truth values (for P,Q,R). One example of such is in row 1 of the third table shown above. Not all rows need to have T's in them. All we need is one row with nothing but T's.
For further reading, check out this page
http://www.butte.edu/resources/interim/wmwu/iLogic/3.2/iLogic_3_2.html
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