CSCI 301 L04 Worksheet

Lecture 4 - Exercises

Part A - Statements

Decide whether or not each of the following is a statement. If it is, say if it is true or false, if possible.

1. Every real number is an even integer.
2. Sets \(\mathbb{Z}\) and \(\mathbb{N}\) are infinite.
3. Either \(x\) is a multiple of \(8\), or it is not.
4. If the integer \(a\) is a multiple of 6, then the integer \(a\) is even.

Part B - Logical Operators (and, or, not)

Let:

• \(N =\) True
• \(P =\) True
• \(Q =\) False
• \(S =\) True

and evaluate the following statements. You may find it helpful to draw a truth table for the more complicated expressions:

1. \(N \land P\)
2. \(Q \land S\)
3. \(Q \lor N\)
4. \(\neg P\)
5. \(P \land (Q \lor S)\)
6. \(N \land P \land Q\)
7. \(\neg N \lor \neg Q\)
8. \((N \lor P) \land \neg(Q \land S)\)

Part C - Conditional Statements

1. Given \(N, P, Q,\) and \(S\) from above, evaluate the following statements:
   1. \(P \Rightarrow Q\)
   2. \(P \Rightarrow S\)
   3. \(Q \Rightarrow S\)
   4. \((N \land P) \Rightarrow (Q \lor S)\)
2. Give the truth value of each of the following conditional statements:
   1. If the moon is made of green cheese, then 1 = 2.
   2. If 1 = 1, then the moon is made of green cheese.
   3. If 1 = 1, then 30 = 30.
   4. If 1 = 1, then 30 = 12.
   5. If 1 = 2, then 30 = 30.
   6. If 1 = 2, then 30 = 10.

Part D - Conditionals in Natural Language

1. Without changing their meanings, convert each of the following sentences into a sentence having the form “If P, then Q”:
   1. Whenever people agree with me I feel I must be wrong.
   2. An integer is divisible by 8 only if it is divisible by 4.
   3. You fail only if you stop writing.

Part E - 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.