Question 263278: 8. Determine which, if any, of the three statements are equivalent
I) If there is no food left on the table, then we have had dinner.
II) If we have had dinner, then there is no food left on the table.
III) If there is food left on the table, then we have not had dinner.
Answer by mducky2(62)   (Show Source):
You can put this solution on YOUR website! In logic, A → B is equivalent to ~B → ~A. Each one is the contrapositive of the other, where both letters are negated (~A becomes A, or A becomes ~A) and reversed (the second letter goes in front of the first). We're going to convert these sentences to equations, then try to find which one is a contrapositive of the first sentence.
First, let's label each concept:
food left on table = F
we have had dinner = D
Next, let's convert the first statement into an equation. The first half of the first sentence says "no food left on the table", so it becomes "not F", or ~F. It ends with "we have had dinner", so that is abbreviated as D. The first sentence converts to:
~F → D
Using the same technique, let's convert the second statement into an equation. The first half of the second sentence says "we have had dinner", so it becomes D. The second half says "no food left on the table", so that becomes ~F. The second sentences converts to:
D → ~F
Finally, let's convert the last statement into an equation. The first half of the third sentence says "food left on the table", so it becomes F. It ends with "we have not had dinner", so that becomes ~D. The third sentences converts to:
F → ~D
Now let's try to figure out which one is the contrapositive. We can see that the first and second sentences are the reverse of each other, but neither one is negated.
~F → D
D → ~F
We can see that the first and third sentences are the negation of each other, but the letters are not reversed.
~F → D
F → ~D
We can see that the second and third sentences are both negated and reversed.
D → ~F
F → ~D
The second and third sentences are contrapositives, so they must be equivalent.
|
|
|
