Math Problem Statement
Show that (PVQ)A] (1PA (QVR)) v ([PA]Q) v ([PA]R) is a tautology
Solution
To show that the expression:
is a tautology, we need to demonstrate that it is always true, regardless of the truth values of , , and . A logical statement is a tautology if its truth table evaluates to true for all possible combinations of truth values of the variables involved.
We will construct the truth table step by step.
1. Breakdown of Components
2. Construction of the Truth Table
| T | T | T | T | T | F | F | F | F | F |
| T | T | F | T | T | F | F | F | F | F |
| T | F | T | T | T | F | F | F | F | F |
| T | F | F | T | F | F | F | F | F | F |
| F | T | T | T | T | T | T | T | F | T |
| F | T | F | T | T | T | T | T | F | T |
| F | F | T | F | T | T | T | F | T | T |
| F | F | F | F | F | T | F | F | F | F |
3. Evaluation of the Final Expression
As we can see from the final column of the truth table, the expression evaluates to True in all cases where is false. This confirms that the logical statement is not a tautology because there are combinations of truth values where the expression evaluates to false.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Propositional Logic
Logical Operators
Tautologies
Formulas
Truth table for evaluating logical expressions
Theorems
Tautology definition
Suitable Grade Level
Grades 11-12 (Advanced Logic)