Question 1): Pei Ann has been dealt two cards from a standard 52 card deck. She holds one in her left hand and one in her right.
Let $p$ be the proposition "The card in Pei Ann's left hand is an ace".
Let $q$ be the proposition "The card in Pei Ann's right hand is an ace"
Let $r$ be the proposition "The card in Pei Ann's left hand is a club".
Let $s$ be the proposition "The card in Pei Ann's right hand is a club".
Write propositions (using just $p, q, r, s$ and logical connectives) corresponding to the following sentences.
- Pei Ann doesn't have two aces.
- Pei Ann has at least one club.
- Pei Ann has the ace of clubs and another club.
Question :2) Show that $(\neg p \lor \neg q) \rightarrow \neg q \equiv p \lor \neg q$ using the laws of logic.
These are two of the questions for my quiz that I had trouble doing.
For question number (1) I am not sure what a standard 52 card deck is, not a cards fan.
I could solve it for the first sentence "Pei Ann doesn't have two aces" I think. I came with $\neg (p \land q)$. Is it correct? For the second sentence I am not exactly sure. Could it be $(p \rightarrow r) \lor (q \rightarrow s)$?
For the third sentence I have no idea how to even begin.
For question number (2) I got to this result $\neg q \lor p \land q$ after applying various logic laws. I could not get it to $p \lor \neg q$.
Any help will be appreciated.
Best Answer
Q1.
-(sentence 1) Pei Ann doesn't have two aces. : $\neg p \land \neg q$
-(sentence 2) Pei Ann has at least one club. : $r \lor s$
-(sentecne 3) Pei Ann has the ace of clubs and another club. : $ (p \land s) \lor (q \land r)$
Q2.
show that (¬p∨¬q)→ ¬q ≡ p∨¬q
(¬p∨¬q)→ ¬q ≡ ¬(p $\land$q) → ¬q ≡ (p $\land$q) $\lor$ $\neg q$ ≡ (p $\lor$ ¬q) $\land$ (q $\lor$ ¬q) ≡ (p $\lor$ ¬q)