Questions: Introduction to Truth Tables

Question 1

How many rows does a truth table for an expression with three variables (P, Q, R) have?

• A. 3
• B. 6
• C. 8
• D. 9

Answer: (C)

Each variable can be true or false (2 values), and with 3 independent variables, the total number of combinations is 2³ = 8. In general, n variables require 2ⁿ rows. Two variables need 4 rows, three need 8, four need 16, and so on.

Question 2

In a truth table for P → Q, the result is FALSE when P is true and Q is false.

• T. True
• F. False

Answer: True

P → Q is false only in the single case where P is true and Q is false. In all other combinations (P true and Q true, P false and Q true, P false and Q false), P → Q is true. This is the defining property of the conditional — it can only be broken when the hypothesis holds but the conclusion does not.

Question 3

Build the truth table for P AND (NOT Q) and identify in which row(s) the expression is true.

Model answer: P=T, Q=T → NOT Q=F → P AND F = F. P=T, Q=F → NOT Q=T → P AND T = T. P=F, Q=T → NOT Q=F → F AND F = F. P=F, Q=F → NOT Q=T → F AND T = F. The expression is true only when P is true and Q is false (row 2).

Building the table step by step: first compute NOT Q for each row, then compute P AND (NOT Q). The expression P ∧ ¬Q is true exactly when P is true and Q is false. Interestingly, this is exactly the condition that makes P → Q false — so P ∧ ¬Q is the negation of P → Q.