Question

Find the expression tree for the formula $$ ((p \rightarrow \neg q) \vee q) \rightarrow q $$ Evaluate the expression tree if proposition \(p\) is \(F\) and proposition \(q\) is \(T\).

Short answer

The expression tree evaluates to True.

Step-by-step solution

Breaking Down the Formula

The given formula is \((p \rightarrow eg q) \vee q) \rightarrow q\). We are going to break it into smaller components and determine the operations at each step. The main operation of the entire formula is an implication \(\rightarrow\) with \(((p \rightarrow eg q) \vee q)\) as the antecedent and \(q\) as the consequent.

Building the Inner Expression Trees

First, let's consider the expression \((p \rightarrow eg q)\). This part of the formula is an implication where \(p\) is the antecedent and \(eg q\) is the consequent. The operation here is an implication \(\rightarrow\). Next, build another tree for \((p \rightarrow eg q) \vee q\). This is a disjunction tree where the left branch is \((p \rightarrow eg q)\) and the right branch is \(q\).

Constructing the Full Expression Tree

Now, integrate the parts into a full expression tree. The top node will be an implication \(\rightarrow\), connecting the disjunction \((p \rightarrow eg q) \vee q\) on the left to \(q\) on the right. This creates a tree structure that represents the full operation of the formula.

Evaluate Base Propositions

Given that \(p = F\) (false) and \(q = T\) (true), let's substitute these into our tree. Start with evaluating the innermost operations: \(eg q = eg T = F\).

Evaluate Inner Implication

Evaluate the expression \(p \rightarrow eg q\) where \(p = F\) and \(eg q = F\). By the definition of implication, \(F \rightarrow F = T\).

Evaluate Inner Disjunction

Next, evaluate \((p \rightarrow eg q) \vee q\), where earlier we found \(p \rightarrow eg q = T\) and \(q = T\). Performing the disjunction: \(T \vee T = T\).

Evaluate the Final Implication

Finally, evaluate the top-level implication \(((p \rightarrow eg q) \vee q) \rightarrow q\), where we've determined both parts: \((p \rightarrow eg q) \vee q = T\) and \(q = T\). By the implication rule, \(T \rightarrow T = T\).

Key concepts

Logical Operators

Logical operators in propositional logic are symbols or words used to connect statements and express logical relationships. They are fundamental building blocks for creating logical expressions. In propositional logic, these operators allow us to define complex statements based on simpler ones.

Here are some basic logical operators:

• Conjunction (\(\land\)): This operator corresponds to "and" in everyday language. A conjunction is true if and only if both operands are true.
• Disjunction (\(\lor\)): This can be thought of as "or". A disjunction is true if at least one operand is true.
• Negation (\(eg\)): This operator negates or inverts the truth value of a statement. If a statement is true, its negation is false, and vice versa.
• Implication (\(\rightarrow\)): Often read as "implies," an implication \(p \rightarrow q\) is false only when \(p\) is true and \(q\) is false.

Understanding these operators is essential to analyze and construct logic-based statements, as seen in expressions like \(((p \rightarrow eg q) \lor q) \rightarrow q\). By applying these operators, you can dissect and evaluate any logical expression.

Propositional Logic

Propositional logic, also known as Boolean logic, deals with statements that are either true or false. These statements, referred to as propositions, can be connected using logical operators to form more complex expressions.

In propositional logic:

• A proposition is a declarative statement with a definite truth value (true or false).
• Atomic propositions are basic expressions without any logical operators applied.
• Composite propositions arise from linking atomic propositions with logical operators.

When working with propositional logic, each proposition can be evaluated based on its structure and the truth values assigned to its components. In our exercise, propositions such as \(p\) and \(q\) were evaluated under some conditions (e.g., \(p = F\), \(q = T\)) to determine the truth value of an entire expression tree. Understanding the atomic and composite propositions allows us to systematically evaluate logical expressions.

Evaluation of Logical Expressions

To evaluate logical expressions, especially those represented in expression trees, involves systematically determining the truth value of composite logical expressions based on given propositions and logical operators.

Here’s how you can approach this:

• Break Down the Expression: Decompose the expression into smaller parts. This allows for easier management and evaluation. In our given formula, separate operations like \(p \rightarrow eg q\) need to be evaluated first.
• Evaluate From the Ground Up: Start with the innermost sub-expressions, moving outwards. First, compute \(eg q\) by inverting \(q\)'s truth value. Then, evaluate \(p \rightarrow eg q\) using known truth values.
• Apply Logical Operators: Begin with basic operations and use the results to solve successive parts of the expression. After completing the inner operations, address larger structures like disjunctions (\((p \rightarrow eg q) \vee q\)) and finally top-level implications (\(((p \rightarrow eg q) \lor q) \rightarrow q\)).
• Utilize Truth Values: Substitute truth values into the expression to systematically solve and confirm the result.

By breaking down and evaluating the given logical expression step-by-step, we can accurately determine the overall truth value, ensuring a comprehensive understanding of how various components contribute to the final result.