Overview
This lecture explains how to represent the exclusive or (XOR) using standard logical notation and verifies it with a truth table.
Exclusive Or (XOR) Definition and Expression
- Exclusive or (XOR) is true when one component is true and the other is false.
- Symbolically, XOR can be expressed as: (P or Q) and not (P and Q).
- This means the statement is true if either P or Q is true, but not both.
Constructing and Verifying the Truth Table
- To verify the expression, build a truth table with all combinations of P and Q.
- List columns for P, Q, P or Q, P and Q, not (P and Q), and the full expression.
- Fill the "P or Q" column: true when at least one is true, false only if both are false.
- Fill the "P and Q" column: true only when both are true, otherwise false.
- Negate "P and Q" for "not (P and Q)"; true when "P and Q" is false, and vice versa.
- Combine "P or Q" and "not (P and Q)" with 'and'; true only when both are true (i.e., one of P or Q is true, but not both).
- The resulting column is true only in rows where exactly one of P or Q is true.
Key Terms & Definitions
- Exclusive Or (XOR) — A logical operation true when exactly one input is true.
- Disjunction (or) — True if at least one input is true.
- Conjunction (and) — True only if both inputs are true.
- Negation (not) — Flips the truth value: true becomes false, false becomes true.
- Truth Table — A table showing all possible truth values for logical statements.
Action Items / Next Steps
- Practice constructing truth tables for compound logical statements.
- Review the steps for expressing logical connectives with basic operators.
- Prepare for the next lecture on advanced truth table concepts.