CSCI 301 L07 Worksheet

Lecture 7 - Exercises

Part A - Biconditional Statements

1. Derive the truth table for \(P \Leftrightarrow Q\) by filling out a truth table with the following five columns:

• \(P\)
• \(Q\)
• \(P\Rightarrow Q\)
• \(Q\Rightarrow P\)
• \(P \Leftrightarrow Q\)

Part B - Truth Table Exercise

Three men, Alfred, Kurt, and Rudolf are accused of a crime. They give the following testimony:

• Alfred: Kurt is guilty and Rudolf is innocent.

2. Kurt: If Alfred is guilty, then so is Rudolf.
3. Rudolf: I am innocent but at least one of the others is guilty.

Define the following statements:

• A = “Alfred is innocent”
• B = “Kurt is innocent”
• R = “Rudolf is innocent”

1. Fill in the top row of the truth table by converting each suspect’s testimony into symbols.
2. Fill in each row of the truth table based on the values in the \(A, K, R\) columns.
3. Can everyone’s testimony be true?
4. Assuming everyone is innocent, who committed perjury?
5. Is it possible that everyone who was innocent told the truth, and everyone who was guilty was lying? If so, who is innocent in this case?

Part C - Equivalences

1. Verify using a truth table that \(P \Leftrightarrow Q\) is logically equivalent to \((P \land Q) \lor (\neg P \land \neg Q)\)
2. Find a way to express \(P \Rightarrow Q\) without using \(\Rightarrow\).
3. Find a way to express \(\neg (P \land Q)\) without using \(\land\) or \(\Rightarrow\).
4. Verify that \(P \Rightarrow Q\) is equivalent to \((\neg Q) \Rightarrow (\neg P)\).

Part D - Quantifiers

1. Translate each of the following English statements into symbolic form.
   1. All natural numbers are integers.
   2. Every integer is even.
   3. Every integer that is not even is odd.
   4. For every integer x, there is an integer y for which x + y = 0.
2. Translate each of the following into English:
   1. \(\forall x \in \mathbb{R}, x^2 > 0\)
   2. \(\exists a \in \mathbb{R}, \forall x \in \mathbb{R}, ax = x\)
   3. \(\forall n \in \mathbb{N}, \exists X \in \mathcal{P}(\mathbb{N}), |X| < n\)