CSCI 301 L05 Worksheet

Lecture 5 - Exercises

Part A - Biconditional Statements

1. Fill out a truth table with the following five columns:

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

Describe in words under what circumstances \(P \Leftrightarrow Q\) is true.

Part B - Truth Table Exercise

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

1. 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 - Quantifiers

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