CSCI 301 L06 Worksheet

Lecture 6 - Exercises

Part A - 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. Verify using a truth table that \(P \Rightarrow Q\) is equivalent to \((\neg Q) \Rightarrow (\neg P)\).
3. Find a way to express \(P \Rightarrow Q\) without using \(\Rightarrow\).
4. Find a way to express \(\neg (P \land Q)\) without using \(\land\) or \(\Rightarrow\).
5. Verify using a truth table that \(\neg(P \lor Q) \equiv (\neg P) \land (\neg Q)\)
6. Verify using a truth table that \(\neg(P \Rightarrow Q) \equiv P \land \neg Q\)

Part B - Negating Statements

For each of the following, convert it to symbols, negate it, simplify as much as possible, then translate it back into English.

1. \(x\) is positive, but \(y\) is not positive
2. Every even integer greater than 2 is the sum of two primes.
3. At least one of the integers \(a\) and \(b\) is odd.
4. \(2a\) is even if and only if \(a\) is an integer
5. There exists a real number \(y\) for which \(x < y\) for any real number \(x\).