Math Problem Statement

Construct a truth table for (p ∨ q) ∨ (~p ∨ ~r). Use T for true and F for false.

Solution

The expression you provided is:

(pq)(¬p¬r)(p \lor q) \lor (\neg p \lor \neg r)

where:

  • pp and qq represent statements,
  • \lor is the logical OR,
  • ¬\neg is the logical NOT.

We'll construct a truth table for this expression. Since there are three variables pp, qq, and rr, the truth table will have 23=82^3 = 8 rows, covering all possible truth values for pp, qq, and rr.

Here is the step-by-step breakdown:

  1. Compute pqp \lor q.
  2. Compute ¬p\neg p.
  3. Compute ¬r\neg r.
  4. Compute ¬p¬r\neg p \lor \neg r.
  5. Finally, compute the full expression (pq)(¬p¬r)(p \lor q) \lor (\neg p \lor \neg r).

Let me now construct the full truth table.

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Math Problem Analysis

Mathematical Concepts

Logic
Truth Table
Propositional Logic

Formulas

Logical OR ( ∨ )
Logical NOT ( ~ )

Theorems

De Morgan's Laws
Basic Truth Table Construction

Suitable Grade Level

Grades 9-12