Discrete Math Quiz: Truth Tables

What this quiz covers

This quiz focuses on Truth Tables, giving you a quick way to practice the rules, question types, and explanations that matter most for Discrete Math.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

Question 1

1. $\neg(p \lor q)$, which is the NOR operation between $p$ and $q$
2. $\neg(p \land q)$, which is the NAND operation between $p$ and $q$ (correct answer)
3. $p \oplus q$, which is the exclusive OR operation between $p$ and $q$
4. $p \leftrightarrow q$, which is the biconditional operation between $p$ and $q$

Question 2

1. Exactly 4 assignments: when any two variables have the same truth value
2. Exactly 2 assignments: when all three variables are true or all are false (correct answer)
3. Exactly 6 assignments: when at least two variables have the same truth value
4. Exactly 3 assignments: when exactly two variables have the same truth value

Question 3

1. When $p$ and $q$ are both false, and either $r$ or $s$ is true
2. When at least one of $p$ or $q$ is true, and at least one of $r$ or $s$ is true (correct answer)
3. When $p$ and $q$ are both true, and $r$ and $s$ are both false
4. When exactly one of $p$ or $q$ is true, and exactly one of $r$ or $s$ is false

Question 4

1. This expression is a tautology (always true)
2. This expression is logically equivalent to $P \oplus Q$
3. This expression is satisfiable but not a tautology
4. This expression is a contradiction (always false) (correct answer)

Question 5

1. Exactly two rows show the expression as false
2. Exactly one row shows the expression as false
3. No rows show the expression as false (correct answer)
4. Exactly three rows show the expression as false

Question 6

1. Exactly five rows evaluate to true
2. Exactly four rows evaluate to true
3. Exactly three rows evaluate to true (correct answer)
4. Exactly six rows evaluate to true

Question 7

1. $P = \text{True}, Q = \text{True}$
2. $P = \text{False}, Q = \text{False}$
3. $P = \text{False}, Q = \text{True}$
4. $P = \text{True}, Q = \text{False}$ (correct answer)

Question 8

1. De Morgan's laws applied to conditional statements (though these concern negating conjunctions/disjunctions)
2. The law of excluded middle in classical logic (though this involves different operators)
3. The law of contraposition (creating a contradiction since contraposition is always valid) (correct answer)
4. The distributive property of conjunction over disjunction (though this involves different logical structures)

Question 9

1. $\neg p \lor \neg q \lor \neg r \lor s$, representing a disjunction of literals
2. $(p \land q \land r) \rightarrow s$, representing a single conditional with compound antecedent
3. $\neg(p \land q \land r \land \neg s)$, representing the negation of a specific conjunction
4. All of the above expressions are logically equivalent to the original nested conditional (correct answer)

Question 10

1. The expression is always true, forming a tautology in all cases
2. The expression is true exactly when $p$ and $q$ have the same truth values
3. The expression is always false, forming a contradiction in all cases
4. The expression is true exactly when $p$ and $q$ have different truth values (correct answer)

Question 11

1. $p$ is true, $q$ is true, $r$ is true, and $s$ is false (correct answer)
2. $p$ is false, $q$ is false, $r$ is false, and $s$ is true
3. $p$ is true, $q$ is false, $r$ is true, and $s$ is false
4. $p$ is false, $q$ is true, $r$ is false, and $s$ is true

Question 12

1. $p \lor s$ must be true in all valid cases
2. $p \lor s$ must be false in all valid cases
3. $p \lor s$ could be either true or false depending on other values (correct answer)
4. $p \lor s$ is true when $r$ is true and false when $r$ is false

Question 13

1. $s$ and $t$ can have any truth values, depending on $r$
2. At least one of $s$ or $t$ must be true, but not necessarily both
3. Both $s$ and $t$ must be true, regardless of other variable values (correct answer)
4. $s$ and $t$ must be true only if $q$ is also true

Question 14

1. The student is correct; this is a tautology by De Morgan's law (correct answer)
2. The student is incorrect; this is only true when $p$ and $q$ are both false
3. The student is incorrect; this is a contradiction that's always false
4. The student is incorrect; this is contingent, being false only when $p$ and $q$ are both true

Question 15

1. 2 rows: when all variables have the same truth value (correct answer)
2. 4 rows: when any two variables have the same truth value
3. 6 rows: when at least two variables have the same truth value
4. 3 rows: when exactly two variables have the same truth value

Question 16

1. The expression is false in exactly zero rows (correct answer)
2. The expression is false in exactly one row
3. The expression is false in exactly two rows
4. The expression is false in exactly three rows

Question 17

1. $P = \text{True}, Q = \text{True}, R = \text{False}$
2. $P = \text{True}, Q = \text{False}, R = \text{False}$ (correct answer)
3. $P = \text{False}, Q = \text{True}, R = \text{False}$
4. $P = \text{False}, Q = \text{False}, R = \text{True}$

Question 18

1. When $P$ and $Q$ have the same truth values
2. When $P$ and $Q$ have different truth values (correct answer)
3. When at least one of $P$ or $Q$ is false
4. When both $P$ and $Q$ are true

Question 19

1. Both $P$ and $Q$ must be false, and $R$ must be false (correct answer)
2. $P$ must be false, $Q$ can be either true or false, and $R$ must be false
3. $P$ can be either true or false, $Q$ must be false, and $R$ must be false
4. $P$ must be true, $Q$ must be false, and $R$ must be false