Translation of sentence to logic formula

Question

Here are four sentences:

1. If Jessy moves his truck, Irene will play her guitar
2. Irene will only move her car, if Jessy moves his garbage cans
3. It is not the case, that Jessy will move his carbage cans and not move his truck
4. If Jessy moves his garbage cans, Irene will move her car and play the guitar

Suppose:

• t = "Jessy moves his truck"
• g = "Irene will play her guitar"
• c = "Irene will move her car"
• gc = "Jessy moves his garbage cans"

I have translated the sentences as stated below:

1. $t \rightarrow g$
2. $gc \rightarrow c$
3. $\neg (gc \land \neg t)$
4. $t\rightarrow (c \land g) $

But I'm very unsure about the third sentence. Have I done this right?

Answer 1

The third one is fine, but the second one is wrong. It says ‘Irene will only move her car, if Jessy moves his garbage cans’, but your proposed translation, $gc\to c$, admits the possibility that Irene will move her car and Jessy will not move his garbage cans. I suggest you make a truth table for this sentence and try to extract the translation from the truth table.