Complete Guide to Mastering Logic and Truth Tables

Step-by-step Guide: Logic and Truth Tables

Basic Logical Statements
Logical statements are assertions that can be either true or false, but not both. We typically use variables like \( p \) or \( q \) to denote these statements. For instance, \( p \) could stand for “It is raining,” and \( q \) could represent “I will carry an umbrella.”

Logical Operators
Logical operators are used to form compound statements from basic ones. Here are the primary logical operators:

• Conjunction (AND) denoted by \(\land\): If both \( p \) and \( q \) are true, then \( p \land q \) is true.
• Disjunction (OR) denoted by \(\lor\): If either \( p \) or \( q \) is true, then \( p \lor q \) is true.
• Negation (NOT) denoted by \(\neg\): If \( p \) is true, then \(\neg p \) is false and vice versa.
• Conditional (IF…THEN) denoted by \(\rightarrow\): “If \( p \) then \( q \)” is false only when \( p \) is true and \( q \) is false.
• Biconditional (IF AND ONLY IF) denoted by \(\leftrightarrow\): True if both \( p \) and \( q \) have the same truth value.

Constructing a Truth Table
A truth table lists all possible truth values for a given logical statement or compound statement. It provides a systematic way to evaluate the truth or falsity of a statement based on the truth values of its components.

Here’s how to construct one:

• List all possible combinations of truth values for the statements involved.
• Evaluate the compound statement for each combination.

Examples

Example 1:
Construct the truth table for the logical statement \( p \lor \neg p \):

This table shows that \( p \lor \neg p \) is always true, which makes it a tautology.

Example 2:
Construct the truth table for the conditional statement \( p \rightarrow q \):

Note that the only time the conditional statement is false is when \( p \) is true and \( q \) is false.

Practice Questions:

1. Construct the truth table for the statement \( \neg p \land q \).
2. Evaluate the truth table for the biconditional statement \( p \leftrightarrow q \).
3. Determine the truth value of the statement \( (p \land q) \lor \neg p \) when \( p \) is true and \( q \) is false.

Answers:

1. Truth Table for \( \neg p \land q \)

   \( p \)	\( q \)	\( \neg p \land q \)
   T	T	F
   T	F	F
   F	T	T
   F	F	F

2. Truth Table for \( p \leftrightarrow q \)

   \( p \)	\( q \)	\( p \leftrightarrow q \)
   T	T	T
   T	F	F
   F	T	F
   F	F	T

3. The truth value of \( (p \land q) \lor \neg p \) when \( p \) is true and \( q \) is false is True.