Question

Read the statement shown below.

If Amelia finishes her homework, then she will go to the park.

Which of these is logically equivalent to the given statement? (1 point)


1. If Amelia did not go to the park, then she did not finish her homework.
2. If Amelia did not finish her homework, then she will go to the park.
3. If Amelia goes to the park, then she did not finish her homework.
4. If Amelia finishes her homework, then she cannot go to the park.

Answer 1

Number 1 is the same thing but told differently.

Answer 2

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.

==========

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 :)