Hello there.
In this problem, we can use our intuition of logic, but I will show a proof of the result in a truth table later. Then, let's get started!
Given:
→ Amelia finishes the homework (sentence H, can be True or False)
→ Amelia goes to the park (P, true or false)
Then, we have: If H, then P. Logically:
H ⇒ P
Then we can think: everytime she does the homework, she goes to the park. Therefore, if she did not go to the park, she will not have finished the homework (It is an equivalent sentence).
Alternative 1.
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Now, let's prove that (H ⇒P) is equivalent to (¬P ⇒ ¬H), via the truth table:
H P ¬H ¬P (H ⇒ P) (¬P ⇒ ¬H)
T T F F T T
T F F T F F
F T T F T T
F F T T T T
As we can see, the results are identical, therefore, the sentences are indeed equivalent.
I hope it hepls :)
