Truth Tables Demonstrating Logical Equivalence Of Statements P, Q, And R
In the realm of mathematical logic, understanding the equivalence of statements is paramount. Logical equivalence signifies that two statements possess the same truth values under all possible circumstances. To rigorously demonstrate this equivalence, we often employ truth tables, a systematic method for evaluating the truth values of compound statements. This article delves into demonstrating the logical equivalence of statement pairs using truth tables, focusing on the given statements involving propositional variables p, q, and r. We will construct truth tables to compare the truth values of the statement pairs: (a) $(p ext{ and } q) ext{ if and only if } p$ and $p ext{ implies } q$, and (b) $p$ and $(p ext{ or } q) ext{ and } (p ext{ or not } q)$. Through meticulous analysis of these tables, we aim to establish the logical equivalence of each pair, providing a comprehensive understanding of their interrelation.
a. Logical Equivalence of $(p ext{ and } q) ext{ if and only if } p$ and $p ext{ implies } q$
To demonstrate the logical equivalence of the statements $(p ext{ and } q) ext{ if and only if } p$ and $p ext{ implies } q$, we will construct a truth table that systematically evaluates the truth values of both statements for all possible combinations of truth values for p and q. The truth table will have columns for p, q, the conjunction $(p ext{ and } q)$, the biconditional $(p ext{ and } q) ext{ if and only if } p$, and the conditional $p ext{ implies } q$. By comparing the truth values in the columns corresponding to the biconditional and the conditional, we can determine whether the two statements are logically equivalent.
Let's begin by constructing the truth table. The first two columns will represent the truth values of p and q, which can be either true (T) or false (F). The conjunction $(p ext{ and } q)$ is true only when both p and q are true, and false otherwise. The biconditional $(p ext{ and } q) ext{ if and only if } p$ is true when both $(p ext{ and } q)$ and p have the same truth value, and false otherwise. Finally, the conditional $p ext{ implies } q$ is false only when p is true and q is false, and true in all other cases.
The truth table is as follows:
| p | q | p and q | (p and q) if and only if p | p implies q |
|---|---|---|---|---|
| T | T | T | T | T |
| T | F | F | F | F |
| F | T | F | T | T |
| F | F | F | T | T |
By examining the columns for $(p ext{ and } q) ext{ if and only if } p$ and $p ext{ implies } q$, we observe that the truth values are not identical in all rows. Specifically, in the second row where p is true and q is false, the biconditional is false, while the conditional is also false. However, in the remaining rows, the truth values match. Therefore, the statements $(p ext{ and } q) ext{ if and only if } p$ and $p ext{ implies } q$ are not logically equivalent.
Detailed Analysis of the Truth Table
To gain a deeper understanding of why the two statements are not logically equivalent, let's analyze each row of the truth table in detail.
- Row 1 (p = T, q = T): In this case, both p and q are true. Therefore, $(p ext{ and } q)$ is also true. The biconditional $(p ext{ and } q) ext{ if and only if } p$ is true because both $(p ext{ and } q)$ and p are true. The conditional $p ext{ implies } q$ is true because p is true and q is true.
- Row 2 (p = T, q = F): Here, p is true and q is false. The conjunction $(p ext{ and } q)$ is false because q is false. The biconditional $(p ext{ and } q) ext{ if and only if } p$ is false because $(p ext{ and } q)$ is false and p is true. The conditional $p ext{ implies } q$ is false because p is true and q is false.
- Row 3 (p = F, q = T): In this scenario, p is false and q is true. The conjunction $(p ext{ and } q)$ is false because p is false. The biconditional $(p ext{ and } q) ext{ if and only if } p$ is true because both $(p ext{ and } q)$ and p are false. The conditional $p ext{ implies } q$ is true because p is false.
- Row 4 (p = F, q = F): In this case, both p and q are false. The conjunction $(p ext{ and } q)$ is false because both p and q are false. The biconditional $(p ext{ and } q) ext{ if and only if } p$ is true because both $(p ext{ and } q)$ and p are false. The conditional $p ext{ implies } q$ is true because p is false.
The analysis confirms that the truth values of the biconditional and the conditional differ in at least one row, specifically when p is true and q is false. This discrepancy demonstrates that the statements $(p ext{ and } q) ext{ if and only if } p$ and $p ext{ implies } q$ are not logically equivalent.
b. Logical Equivalence of $p$ and $(p ext{ or } q) ext{ and } (p ext{ or not } q)$
To determine the logical equivalence of $p$ and $(p ext{ or } q) ext{ and } (p ext{ or not } q)$, we will again employ a truth table. This truth table will encompass columns for p, q, the disjunction $(p ext{ or } q)$, the negation of q ($ ext{not } q$), the disjunction $(p ext{ or not } q)$, and the conjunction $(p ext{ or } q) ext{ and } (p ext{ or not } q)$. By comparing the truth values of p and the final conjunction, we can ascertain their logical equivalence.
The construction of the truth table begins with all possible combinations of truth values for p and q. The disjunction $(p ext{ or } q)$ is true if either p or q (or both) is true, and false only when both are false. The negation $ ext{not } q$ reverses the truth value of q. The disjunction $(p ext{ or not } q)$ is true if either p or $ ext{not } q$ (or both) is true, and false only when both are false. Finally, the conjunction $(p ext{ or } q) ext{ and } (p ext{ or not } q)$ is true only when both disjunctions are true, and false otherwise.
The truth table is presented below:
| p | q | not q | p or q | p or not q | (p or q) and (p or not q) |
|---|---|---|---|---|---|
| T | T | F | T | T | T |
| T | F | T | T | T | T |
| F | T | F | T | F | F |
| F | F | T | F | T | F |
By examining the columns for p and $(p ext{ or } q) ext{ and } (p ext{ or not } q)$, we observe that the truth values are identical in all rows. This confirms that the statements $p$ and $(p ext{ or } q) ext{ and } (p ext{ or not } q)$ are logically equivalent.
Detailed Analysis of the Truth Table
To solidify our understanding, let's analyze each row of the truth table in detail.
- Row 1 (p = T, q = T): In this row, p is true and q is true. Therefore, $ ext{not } q$ is false. The disjunction $(p ext{ or } q)$ is true because p is true. The disjunction $(p ext{ or not } q)$ is true because p is true. The conjunction $(p ext{ or } q) ext{ and } (p ext{ or not } q)$ is true because both disjunctions are true.
- Row 2 (p = T, q = F): Here, p is true and q is false. Thus, $ ext{not } q$ is true. The disjunction $(p ext{ or } q)$ is true because p is true. The disjunction $(p ext{ or not } q)$ is true because both p and $ ext{not } q$ are true. The conjunction $(p ext{ or } q) ext{ and } (p ext{ or not } q)$ is true because both disjunctions are true.
- Row 3 (p = F, q = T): In this case, p is false and q is true. Consequently, $ ext{not } q$ is false. The disjunction $(p ext{ or } q)$ is true because q is true. The disjunction $(p ext{ or not } q)$ is false because both p and $ ext{not } q$ are false. The conjunction $(p ext{ or } q) ext{ and } (p ext{ or not } q)$ is false because one of the disjunctions is false.
- Row 4 (p = F, q = F): Here, both p and q are false. Therefore, $ ext{not } q$ is true. The disjunction $(p ext{ or } q)$ is false because both p and q are false. The disjunction $(p ext{ or not } q)$ is true because $ ext{not } q$ is true. The conjunction $(p ext{ or } q) ext{ and } (p ext{ or not } q)$ is false because one of the disjunctions is false.
The detailed analysis confirms that the truth values of p and $(p ext{ or } q) ext{ and } (p ext{ or not } q)$ are identical across all rows. This definitively demonstrates that the statements $p$ and $(p ext{ or } q) ext{ and } (p ext{ or not } q)$ are logically equivalent.
In this article, we have rigorously examined the logical equivalence of two pairs of statements using truth tables. Our analysis revealed that the statements $(p ext{ and } q) ext{ if and only if } p$ and $p ext{ implies } q$ are not logically equivalent, as their truth values differ in at least one case. Conversely, we demonstrated that the statements $p$ and $(p ext{ or } q) ext{ and } (p ext{ or not } q)$ are indeed logically equivalent, as their truth values coincide across all possible scenarios. These findings underscore the importance of truth tables as a powerful tool for verifying logical equivalences in mathematical logic. Understanding these equivalences is crucial for constructing sound arguments and developing logical reasoning skills. By mastering the use of truth tables, we can effectively analyze and manipulate logical statements, leading to a deeper comprehension of mathematical principles.
