Truth Tables

Show your work for each problem.

1.
How many rows does a truth table with 3 variables have?
1. a)9
2. b)3
3. c)8
4. d)6
Answer: 8
$2^3 = 2 \times 2 \times 2 = 8$ rows. 6 would be $2 \times 3$ , which misses combinations.
2.
How many rows does a truth table for $P$ and $Q$ need?
Answer: 4
$2^2 = 4$ rows: TT, TF, FT and FF.
3.
Set up the variable columns of a truth table for $P$ , $Q$ and $R$ in the standard order.
Answer: 8
1. How many rows does the table need? 8
2. Column $P$ starts with a block of T values. How many T values are in that first block? 4
3. Column $Q$ switches twice as often. How many T values are in its first block? 2
4. How many T values are in the first block of column $R$ ? 1
4.
In the standard order of an 8-row truth table, which pattern does the last column $R$ have?
1. a)F, F, F, F, T, T, T, T
2. b)T, T, F, F, T, T, F, F
3. c)T, F, T, F, T, F, T, F
4. d)T, T, T, T, F, F, F, F
Answer: T, F, T, F, T, F, T, F
The last column alternates in blocks of 1: T, F, T, F, T, F, T, F. Blocks of 4 belong to $P$ and blocks of 2 to $Q$ .
5.
A truth table has 64 rows. How many variables does the expression have?
Answer: 6
$2^6 = 64$ , so the expression has 6 variables (2, 4, 8, 16, 32, 64 is six doublings).
6.
Build the truth table for $\neg P \land Q$ . In which row is it true? Use the standard row order (row 1 is $P$ = T, $Q$ = T).
Answer: 3
1. How many rows does the table have? 4
2. In how many rows is $\neg P$ true? 2
3. Which row number (1 to 4) has both $\neg P$ and $Q$ true? 3
7.
In how many of the 4 rows of the truth table is $\neg P \lor Q$ true?
Answer: 3
$\neg P$ : F, F, T, T and $Q$ : T, F, T, F. OR gives T, F, T, T, so the expression is true in 3 rows and false only when $P$ is true and $Q$ is false.
8.
Which expression is a tautology?
1. a) $P \land Q$
2. b) $P \lor \neg P$
3. c) $\neg P \lor Q$
4. d) $P \land \neg P$
Answer: $P \lor \neg P$
$P \lor \neg P$ is always true: one of $P$ and $\neg P$ is always true. $P \land \neg P$ is always false, and $P \land Q$ and $\neg P \lor Q$ are each false in at least one row.
9.
In how many of the 8 rows is $P \land (Q \lor R)$ true?
Answer: 3
1. In how many of the 8 rows is $Q \lor R$ true? 6
2. In how many rows is $P$ true? 4
3. In how many rows are $P$ and $Q \lor R$ both true? 3
10.
In how many of the 8 rows of the truth table is $P \lor (Q \land R)$ true?
Answer: 5
$P$ is true in rows 1 to 4, which gives 4 true rows. In rows 5 to 8 ( $P$ false), $Q \land R$ is true only in row 5. Total: $4 + 1 = 5$ rows.
11.
Use a truth table to check whether $(P \lor Q) \lor \neg Q$ is a tautology.
Answer: 0
1. In how many of the 4 rows is $P \lor Q$ true? 3
2. In how many rows is $\neg Q$ true? 2
3. In how many rows is $(P \lor Q) \lor \neg Q$ true? 4
4. How many rows are false? (0 false rows means a tautology) 0
12.
Which expression is logically equivalent to $\neg(P \lor Q)$ ?
1. a) $P \land Q$
2. b) $\neg P \lor \neg Q$
3. c) $\neg P \land \neg Q$
4. d) $\neg P \lor Q$
Answer: $\neg P \land \neg Q$
$\neg(P \lor Q)$ : F, F, F, T. $\neg P \land \neg Q$ : F, F, F, T. The columns match in every row (De Morgan). $\neg P \lor \neg Q$ gives F, T, T, T, which is different.
13.
Use a truth table to check whether $(P \land Q) \land \neg P$ is a contradiction. Use the standard row order (row 1 is $P$ = T, $Q$ = T).
Answer: 0
1. Which row number (1 to 4) is the only row where $P \land Q$ is true? 1
2. In how many rows is $\neg P$ true? 2
3. In how many rows is $(P \land Q) \land \neg P$ true? 0
14.
What does the truth table of $(P \lor Q) \land \neg P$ show?
1. a)It is a contradiction
2. b)It is equivalent to $P$
3. c)It is neither: true in exactly 1 row
4. d)It is a tautology
Answer: It is neither: true in exactly 1 row
$P \lor Q$ : T, T, T, F and $\neg P$ : F, F, T, T. AND gives F, F, T, F. It is true in exactly 1 row (row 3), so it is neither a tautology nor a contradiction, and it is not equivalent to $P$ (T, T, F, F).
15.
In how many of the 4 rows is $(P \land \neg Q) \lor (\neg P \land Q)$ true?
Answer: 2
$P \land \neg Q$ is true only in row 2 (T, F) and $\neg P \land Q$ only in row 3 (F, T). OR gives F, T, T, F: 2 true rows. The expression is true exactly when $P$ and $Q$ are different.
16.
Use a truth table to check whether $P \land (Q \lor R)$ and $(P \land Q) \lor (P \land R)$ are equivalent.
Answer: 0
1. How many rows does the table have? 8
2. In how many rows is $P \land (Q \lor R)$ true? 3
3. In how many rows is $(P \land Q) \lor (P \land R)$ true? 3
4. In how many rows do the two final columns differ? 0
17.
A student claims that $\neg(P \land Q)$ is equivalent to $\neg P \land \neg Q$ . Check it with a truth table. Use the standard row order (row 1 is $P$ = T, $Q$ = T).
Answer: 2
1. In how many of the 4 rows is $\neg(P \land Q)$ true? 3
2. In how many rows is $\neg P \land \neg Q$ true? 1
3. In how many rows do the two final columns differ? 2
4. What is the number of the first row where they differ? 2
18.
Build the truth table for $\neg P \lor (Q \land \neg R)$ . In how many of the 8 rows is it true?
Answer: 5
1. In how many of the 8 rows is $\neg P$ true? 4
2. In how many rows is $Q \land \neg R$ true? 2
3. In how many rows is $\neg P \lor (Q \land \neg R)$ true? 5
4. How many rows are false? 3