Propositional Logic

Construct a truth table for the compound statement $((p \to q) \land \neg p) \to \neg q$

Consider the statement about a party, “If it's your birthday or there will be cake, then there will be cake.”

1. Translate the above statement into symbols. Clearly state which statement is $P$ and which is $Q$.

2. Make a truth table for the statement.

3. Assuming the statement is true, what (if anything) can you conclude if there will be cake?

4. Assuming the statement is true, what (if anything) can you conclude if…

Make a truth table for the statement $(p \vee q) \to (p \wedge q)$.

Make a truth table for the statement $\neg p \wedge (q \to p)$. What can you conclude about $p$ and $q$ if you know the statement is true?

State the converse, inverse, and contrapositive of each of the following conditional statements:

1. If it rains today, then I will bring an umbrella.
2. Whenever I drive my car, I do not use my phone.
3. When I stay up too late, it's necessary that I sleep until noon.

A classic example is that we're on the island of knights and knaves. Knights always tell the truth. Knaves always lie.

We encounter two people A and B.

• A says: “B is a knight.”
• B says: “The two of us are opposite types.”

What is the conclusion?