Truth Table And Logical Expression Analysis

Hey guys! Today, we're diving into the fascinating world of logical operators and truth tables. We've got a special operator defined, and we're going to break it down step by step. So, grab your thinking caps, and let's get started!

Defining the Operator *

Alright, so we have a new logical operator $*$ defined as:

$p * q = (q \land \neg p) \rightarrow \neg q$

This means that $p * q$ is logically equivalent to "if ($q$ and not $p$), then not $q$". Understanding this definition is crucial before we start constructing the truth table. The operator combines conjunction ($\land$), negation ($\neg$), and implication ($\rightarrow$) in a specific order. The expression $(q \land \neg p)$ represents the conjunction of $q$ and the negation of $p$, meaning both $q$ and $\neg p$ must be true for the expression to be true. The implication $(q \land \neg p) \rightarrow \neg q$ then states that if $(q \land \neg p)$ is true, then $\neg q$ must also be true. To fully grasp the behavior of this operator, we need to consider all possible combinations of truth values for $p$ and $q$, which is precisely what a truth table helps us visualize. Understanding the operator also involves recognizing the precedence of logical operations: negation is typically evaluated first, followed by conjunction and disjunction, and finally implication and biconditional. This order ensures that the expression is evaluated consistently and unambiguously. Additionally, it's important to note that this operator is not one of the standard logical operators (such as AND, OR, NOT, implication, etc.), which makes it an interesting exercise to analyze its properties and behavior. This analysis will help us determine its truth table and eventually classify the given logical expression involving this operator.

Constructing the Truth Table for $p * q$

Now, let's build the truth table for $p * q$. This table will show all possible combinations of truth values for $p$ and $q$, and the corresponding truth value for $p * q$.

Explanation:

• Column 1 (p): All possible truth values for $p$ (True, True, False, False).
• Column 2 (q): All possible truth values for $q$ (True, False, True, False).
• Column 3 (¬p): The negation of $p$. When $p$ is True, $\neg p$ is False, and vice versa.
• Column 4 (q ∧ ¬p): The conjunction of $q$ and $\neg p$. This is only True when both $q$ and $\neg p$ are True.
• Column 5 (¬q): The negation of $q$. When $q$ is True, $\neg q$ is False, and vice versa.
• Column 6 ((q ∧ ¬p) → ¬q): The implication. This is only False when $(q \land \neg p)$ is True and $\neg q$ is False. Looking at the table, $(q \land \neg p) \rightarrow \neg q$ is only False when $p$ is False and $q$ is True. In all other cases, it's True. The truth table systematically evaluates the expression $p * q$ for all possible combinations of truth values of $p$ and $q$. This comprehensive evaluation allows us to fully understand the behavior of the operator and its relationship to the truth values of its operands. The truth table clearly shows that the value of $p * q$ depends on the specific combination of truth values for $p$ and $q$, highlighting the conditional nature of the defined operator. Furthermore, this truth table will be instrumental in the next step, where we need to analyze a more complex logical expression involving this operator. By having a clear understanding of the truth values for $p * q$, we can substitute them into the larger expression and determine whether it is a tautology, contradiction, or contingency.

Analyzing (q * p) ∨ (¬q)

Next, we need to determine whether $(q * p) \lor (\neg q)$ is a tautology, contradiction, or contingency. A tautology is always True, a contradiction is always False, and a contingency is sometimes True and sometimes False.

First, let's express $q * p$ using the definition:

$q * p = (p \land \neg q) \rightarrow \neg p$

Now, let's create the truth table for $(q * p) \lor (\neg q)$:

Explanation:

• Column 1 (p): All possible truth values for $p$.
• Column 2 (q): All possible truth values for $q$.
• Column 3 (¬p): The negation of $p$.
• Column 4 (¬q): The negation of $q$.
• Column 5 (p ∧ ¬q): The conjunction of $p$ and $\neg q$.
• Column 6 ((p ∧ ¬q) → ¬p): The implication, which represents $q * p$.
• Column 7 (q * p): Result of $(p \land \neg q) \rightarrow \neg p$.
• Column 8 ((q * p) ∨ (¬q)): The disjunction of $(q * p)$ and $\neg q$. This is True if either $(q * p)$ or $\neg q$ (or both) are True.

Looking at the last column, we see that $(q * p) \lor (\neg q)$ is always True. The final column of the truth table, which represents the truth values of the entire expression $(q * p) \lor (\neg q)$, consists entirely of 'T' (True) values. This means that regardless of the truth values of the individual variables $p$ and $q$, the expression as a whole always evaluates to True. This is the defining characteristic of a tautology. In other words, the expression is logically valid and holds true under all possible interpretations. The expression $(q * p) \lor (\neg q)$ is a tautology, which means that it is always true, regardless of the truth values of $p$ and $q$. To further emphasize this point, consider the disjunction operation $(\lor)$. It only requires one of its operands to be true for the entire expression to be true. In this case, either $(q * p)$ is true, or $\neg q$ is true, or both are true. The truth table systematically explores all these possibilities and confirms that at least one of these conditions is always met. Therefore, the expression $(q * p) \lor (\neg q)$ is a tautology. This result highlights the importance of truth tables in logical analysis, as they provide a rigorous and systematic way to determine the logical properties of complex expressions. By constructing and examining the truth table, we can confidently conclude whether an expression is a tautology, contradiction, or contingency, which has significant implications in various fields such as mathematics, computer science, and philosophy.

Conclusion

So, there you have it! We've defined a new logical operator, constructed its truth table, and analyzed a logical expression involving that operator. We found that $(q * p) \lor (\neg q)$ is a tautology. Hope this helps you guys in understanding the topic. Keep up the great work!