Math Problem Statement

Buatlah tabel kebenaran dari pernyataan majemuk berikut: p ⇒ (p ⇒ q)

Solution

The problem in the image is asking to create a truth table for the following compound statement:

p(pq)p \Rightarrow (p \Rightarrow q)

Let's break this down and create the truth table.

Components:

  1. pp: A simple proposition.
  2. qq: Another simple proposition.
  3. pqp \Rightarrow q: This is an implication statement, which is true unless pp is true and qq is false.
  4. p(pq)p \Rightarrow (p \Rightarrow q): This is the main compound statement for which we need to create the truth table.

Truth Table Construction:

We'll need four columns: pp, qq, pqp \Rightarrow q, and p(pq)p \Rightarrow (p \Rightarrow q).

ppqqpqp \Rightarrow qp(pq)p \Rightarrow (p \Rightarrow q)
TTTT
TFFF
FTTT
FFTT

Explanation:

  1. pqp \Rightarrow q: This column shows the result of the implication pqp \Rightarrow q.

    • When p=Tp = T and q=Tq = T, pqp \Rightarrow q is True.
    • When p=Tp = T and q=Fq = F, pqp \Rightarrow q is False.
    • When p=Fp = F, pqp \Rightarrow q is always True, regardless of qq.
  2. p(pq)p \Rightarrow (p \Rightarrow q): The final column shows the main statement's truth value.

    • When p=Tp = T, the statement depends on pqp \Rightarrow q. If pqp \Rightarrow q is true, then p(pq)p \Rightarrow (p \Rightarrow q) is true; otherwise, it's false.
    • When p=Fp = F, p(pq)p \Rightarrow (p \Rightarrow q) is always true, because any implication with a false antecedent is true.

Would you like a more detailed breakdown of the steps? Also, here are five related questions to extend this concept:

  1. What is the truth table for q(pq)q \Rightarrow (p \Rightarrow q)?
  2. How does the truth table of pqp \wedge q compare to pqp \Rightarrow q?
  3. Can you express p(pq)p \Rightarrow (p \Rightarrow q) using logical operators like ¬\neg (NOT) and \vee (OR)?
  4. How does a biconditional p    qp \iff q differ from pqp \Rightarrow q?
  5. What is the difference between the logical operations \wedge (AND) and \Rightarrow (IMPLICATION)?

Tip: When analyzing compound statements in logic, always break them down into smaller sub-statements to simplify the truth table construction process.

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Math Problem Analysis

Mathematical Concepts

Logic
Propositional Logic
Truth Tables

Formulas

Implication (p ⇒ q)

Theorems

Implication Truth Table

Suitable Grade Level

Grades 10-12