Question

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

Short answer

The expression tree's root is \( \wedge \) with two subtrees: left handles negated conjunctions and right handles complex disjunction.

Step-by-step solution

Understand Logical Operators

Identify each logical operator in the formula. Recognize negation (\( eg \)), conjunction (\( \wedge \)), disjunction (\( \vee \)), and biconditional (\( \leftrightarrow \)). These operators follow precedence rules: negation first, conjunction next, disjunction, and lastly, biconditional. Parentheses override precedence.

Identify the Components of the Expression

Break down the formula into smaller logical components: \((eg(p \wedge q))\), \((eg(q \wedge r))\), \((eg(p \leftrightarrow(eg(eg s))))\), and \(((r \wedge s) \vee(eg q))\). These components are separated by the formula's main operations, which in this case are \( \vee \) and \( \wedge \).

Build the Expression Tree from the Inside Out

Start creating the tree by building from the innermost expressions outwards. For instance, begin with \((p \wedge q)\), \((q \wedge r)\), and \((r \wedge s)\) as leaves, and move towards negations and other logical operations.

Construct Subtrees for Intermediate Expressions

Create subtrees for intermediate expressions: for \( eg(p \wedge q) \) and \( eg(q \wedge r) \), the tree will connect \( eg \) with each conjunction subtree. Similarly, for \((eg(eg s))\), connect two negation nodes with \( s \). Then, for \( (r \wedge s) \vee(eg q) \), join its conjunction and negation branches to \( \vee \).

Assemble the Full Expression Tree

Combine the subtrees to form the entire expression tree. The root of the tree is the main conjunction \( \wedge \). Its left child is the disjunction formed by \( eg(p \wedge q) \) and \( eg(q \wedge r) \). Its right child is the disjunction with \( eg(p \leftrightarrow(eg(eg s))) \) and \((r \wedge s) \vee(eg q)\).

Key concepts

Logical Operators

Logical operators are symbols or words used to connect multiple propositions in logical expressions. They decide how propositions are manipulated and interpreted in an expression tree. In logical expressions, the key operators you need to understand are:

• **Negation (\(eg\))** - This operator flips the truth value of a proposition. If a proposition \(p\) is true, then \(eg p\) is false, and vice versa.
• **Conjunction (\(\wedge\))** - This operator tests the truth of two propositions together. \(p \wedge q\) is true if and only if both \(p\) and \(q\) are true.
• **Disjunction (\(\vee\))** - This checks if at least one of two propositions is true. \(p \vee q\) is true if \(p\) is true, \(q\) is true, or both are true.
• **Biconditional (\(\leftrightarrow\))** - This operator states that both propositions are equivalent. \(p \leftrightarrow q\) is true if both \(p\) and \(q\) are either true or false.

Understanding these operators is crucial as they define how expressions should be disentangled in an expression tree.

Logical Components

Logical components are the individual parts or sub-expressions in a complex logical expression. To manage large formulas, it is important to identify distinct logical segments:

• **Sub-Expressions** - These are smaller parts of the formula, often enclosed in parentheses, that need to be evaluated first due to precedence rules.
• **Representation** - Each component is represented in the expression tree, forming nodes with their respective logical operator.
• **Breaking Down** - The original formula \(((eg(p \wedge q)) \vee(eg(q \wedge r))) \wedge((eg(p \leftrightarrow(eg(eg s)))) \vee(((r \wedge s) \vee(eg q))))\) can be split into logical components: \(eg(p \wedge q)\), \(eg(q \wedge r)\), \(eg(p \leftrightarrow(eg(eg s)))\), and \((r \wedge s) \vee(eg q)\).

By breaking down expressions into logical components, constructing an expression tree becomes more manageable.

Precedence Rules

Precedence rules determine the order in which logical operators are applied within an expression. These rules are essential to understand because they can change the meaning of an expression if not followed correctly:

• **Order of Operations** - The usual order begins with negation, followed by conjunction, then disjunction, and finally biconditional.
• **Overriding with Parentheses** - Parentheses are used to explicitly show operations that take precedence. The operations within parentheses are evaluated first, irrespective of their type.
• **Expression Example** - In the given expression, parentheses are used to prioritize operations like \((eg(p \wedge q)) \vee (eg(q \wedge r))\), forcing evaluations that might not align with default precedence rules.

By following precedence rules, you ensure the logical expression is interpreted and solved correctly, leading to the accurate construction of the expression tree.

Tree Construction Process

Constructing an expression tree involves visualizing the logical structure of an expression. Here’s how you can approach building an expression tree:

• **Start with Leaves** - Begin with the simplest components like conjunctions \((p \wedge q)\) at the bottom, considered as leaves of the tree.
• **Building Upward** - Progress to intermediate expressions by adding operators such as negations \((eg(p \wedge q))\). Connect these operations to form branches.
• **Create Subtrees** - Assemble subtrees for expressions like \((r \wedge s) \vee (eg q)\) by joining them under their respective operator.
• **Combine Subtrees** - Link together all subtrees to establish the full structure. The root of this full tree often represents the outermost logical operation, such as the conjunction \(\wedge\) in our example.

This systematic approach ensures that each part of the logical expression is correctly represented and interconnected within the tree structure.