Question

Find the expression tree for the formula $$((((\neg(\neg p)) \wedge(\neg q)) \wedge r) \vee(((\neg(\neg q)) \wedge(\neg r)) \wedge s)) \leftrightarrow(s \rightarrow p)$$

Short answer

The expression tree root is '↔', branching into left '⋁' and right '⋁' subtrees representing the split binary operations.

Step-by-step solution

Understand the Expression

The given expression is \(((((eg(eg p)) \wedge(eg q)) \wedge r) \vee(((eg(eg q)) \wedge(eg r)) \wedge s)) \leftrightarrow(s \rightarrow p)\). We need to build an expression tree, which is a tree representation of the structure of this logical formula.

Identify Top-Level Operator

The top-level operator of the given expression is the biconditional \(\leftrightarrow\). This means we can break the expression into two main sub-expressions: one for the left side and one for the right side of the biconditional.

Divide into Sub-Expressions

The expression transforms into two parts: 1. \(((((eg(eg p)) \wedge(eg q)) \wedge r) \vee(((eg(eg q)) \wedge(eg r)) \wedge s))\)2. \((s \rightarrow p)\)These will be the two branches stemming from the top-level biconditional.

Break Down Left Sub-Expression

The left sub-expression involves a disjunction \(\vee\), breaking it into - \(((eg(eg p)) \wedge(eg q)) \wedge r\) - \(((eg(eg q)) \wedge(eg r)) \wedge s\)Each of these can further be divided, regarding the conjunction \(\wedge\).

Resolve Left Sub-Expression Components

For the left component:\(((eg(eg p)) \wedge(eg q)) \wedge r\) splits into:- Left child: \((eg(eg p)) \wedge(eg q)\)- Right child: \(r\)Further dividing, \((eg(eg p)) \wedge(eg q)\) becomes:- Left child: \(eg(eg p)\)- Right child: \(eg q\)

Resolve Right Component of Left Sub-Expression

For \(((eg(eg q)) \wedge(eg r)) \wedge s\):- Left child: \((eg(eg q)) \wedge(eg r)\)- Right child: \(s\)Splitting further, \((eg(eg q)) \wedge(eg r)\) becomes:- Left child: \(eg(eg q)\)- Right child: \(eg r\)

Right Sub-Expression

Focus on the simplified right sub-expression \((s \rightarrow p)\). Logical implication can be expressed as \(eg s \vee p\). This forms:- Left child: \(eg s\)- Right child: \(p\)

Construct the Expression Tree

- Root: \(\leftrightarrow\) - Left child: \(\vee\) - Left child: \(\wedge\) - Left child: \(\wedge\) - Left child: \(eg\) - Child: \(eg\) - Child: \(p\) - Right child: \(eg q\) - Right child: \(r\) - Right child: \(\wedge\) - Left child: \(\wedge\) - Left child: \(eg\) - Child: \(eg\) - Child: \(q\) - Right child: \(eg r\) - Right child: \(s\) - Right child: \(\vee\) - Left child: \(eg s\) - Right child: \(p\)

Key concepts

Logical Formulas

Logical formulas are expressions composed of logical symbols and variables.They are used to represent logical relationships or statementsin a structured manner. Logical formulas follow specific rulesand use logical operators such as negation, conjunction, disjunction,and implications to build more complex statements.
An example of a logical formula is \( (p \wedge q) \vee (r \rightarrow s) \), where \( \wedge \) is the conjunction operator, \( \vee \) is the disjunction operator, and \( \rightarrow \) is the logical implication operator.
Understanding the components of a logical formula is crucial in constructing expression trees, which visually represent the hierarchy and structureof these formulas.

Biconditional Operator

The biconditional operator, often represented with the symbol \( \leftrightarrow \), is used to indicate a logical equivalence between two statements. A biconditional formula \( A \leftrightarrow B \) states that both statements "A" and "B" are either true together or false together.
This operator is sometimes referred to as "if and only if",signifying that the truth of one side is entirely dependent on the truth of the other.
In the context of expression trees, the biconditional operator is usuallyat the root level when it's the main connective, and it branches outto its two sub-expressions.
For example, in the expression\( ((((eg(eg p)) \wedge (eg q)) \wedge r) \vee(((eg(eg q)) \wedge (eg r)) \wedge s)) \leftrightarrow (s \rightarrow p) \),the biconditional operator divides the formula into two primary components.

Conjunctions and Disjunctions

Conjunctions and disjunctions are logical operations that connect individual propositions. A conjunction, represented by the symbol \( \wedge \), requires both propositions to be true for the entire expression to evaluate as true. For instance, in \( p \wedge q \), both \( p \) and \( q \) must be true.
Meanwhile, a disjunction, represented by \( \vee \), allows either of the propositions to be true for the entire expression to be true.This means in \( p \vee q \), the overall expression is true if either \( p \) or \( q \) is true.
In an expression tree, these operators create branching points,leading to each operand becoming a branch of the tree,and determining the flow of logical evaluation from top to bottom.

Logical Implication

Logical implication is an operator used to express a conditional relationship between two propositions in a logical formula. It is often represented by the arrow symbol \( \rightarrow \). In the expression \( p \rightarrow q \), it states that if \( p \) holds true,then \( q \) must also be true.
However, if \( p \) is false, the implication \( p \rightarrow q \) is automatically true,regardless of the value of \( q \). This might initially seem counterintuitive,but it aligns with the classical logical understanding of implications.
In the expression tree, logical implication often simplifies to a disjunction of the negation of the first term and the second term, represented as\( eg p \vee q \). This way, it maintains consistency and allows the treeto reflect the conditional nature of implication effectively.