Question

Find the expression tree for the following formulas: (a) \(\neg p \wedge(\neg q \vee r)\) (b) \(p \vee(\neg q \wedge \neg r)\) (c) \(((p \vee q) \leftrightarrow r) \leftrightarrow p\) (d) \((\neg q \wedge \neg r) \leftrightarrow(p \rightarrow(q \vee r))\)

Short answer

Construct expression trees by identifying the main operators and organizing subexpressions accordingly.

Step-by-step solution

Exercise Introduction

We are tasked with finding the expression trees for logical formulas. An expression tree visually represents the structure of a logical formula, with operators as internal nodes and operands as leaf nodes. We will consider each subexpression separately.

Construct Expression Tree for (a)

For the formula \( eg p \wedge(eg q \vee r) \):1. Start with the main operator \( \wedge \) as the root.2. Left child of \( \wedge \) is \( eg p \), making \( eg \) the parent of \( p \).3. Right child of \( \wedge \) is \( eg q \vee r \), with root \( \vee \).4. Below \( \vee \), the left child is \( eg q \), where \( eg \) is the parent of \( q \), and the right child is \( r \).

Construct Expression Tree for (b)

For the formula \( p \vee(eg q \wedge eg r) \):1. The root is the main operator \( \vee \).2. The left child is \( p \).3. The right child is the subexpression \( eg q \wedge eg r \) with root \( \wedge \).4. For the \( \wedge \) node, left child is \( eg q \) with \( eg \) as the parent of \( q \), and the right child is \( eg r \) with \( eg \) as the parent of \( r \).

Construct Expression Tree for (c)

For the formula \( ((p \vee q) \leftrightarrow r) \leftrightarrow p \):1. Root with main operator is \( \leftrightarrow \).2. Left child of root is the subexpression \( (p \vee q) \leftrightarrow r \), with \( \leftrightarrow \) as its root.3. The left child of the \( \leftrightarrow \) is \( p \vee q \) with root \( \vee \).4. Under \( \vee \), left child is \( p \) and right child is \( q \).5. The right child of the first \( \leftrightarrow \) is \( r \).6. The right child of the root \( \leftrightarrow \) is \( p \).

Construct Expression Tree for (d)

For the formula \( (eg q \wedge eg r) \leftrightarrow (p \rightarrow(q \vee r)) \):1. The root is \( \leftrightarrow \).2. The left child is \( eg q \wedge eg r \) with root \( \wedge \).3. For \( \wedge \), the left child is \( eg q \), with \( eg \) as the parent of \( q \), and the right child is \( eg r \) with \( eg \) as the parent of \( r \).4. The right child of \( \leftrightarrow \) is \( p \rightarrow (q \vee r) \) with root \( \rightarrow \).5. Beneath \( \rightarrow \), the left child is \( p \) and the right is \( q \vee r \) with root \( \vee \).6. For \( \vee \), left child is \( q \) and right child is \( r \).

Key concepts

Logical Formulas

Logical formulas are fundamental constructs in logic and computer science. They involve combinations of variables and logical operators to express conditions or propositions. In the context of expression trees, formulas like \[ eg p \wedge (eg q \vee r) \]are used as inputs whose structural representation is to be determined by the tree.

These formulas can involve different logical operators such as:

• Negation (\(eg\)), which reverses the truth value of its operand.
• Conjunction (\(\wedge\)), representing an 'and' relationship.
• Disjunction (\(\vee\)), representing an 'or' relationship.
• Implication (\(\rightarrow\)), meaning 'if...then.'
• Biconditional (\(\leftrightarrow\)), expressing equivalence between operands.

The operands in these formulas—typically letters like \(p\), \(q\), \(r\)—are placeholders representing values in logical contexts. Understanding these components is critical to constructing and interpreting expression trees.

Operators and Operands

Operators and operands make up the core elements of logical formulas. Understanding their roles is crucial for anyone working with logical expressions or constructing expression trees.

**Operators** are symbols or keywords that denote the operations to be performed on operands. In logical expressions, operators like \(eg\), \(\wedge\), \(\vee\), \(\rightarrow\), and \(\leftrightarrow\) determine how the truth values of operands combine:

• \(eg\): Unary operator that signifies negation.
• \(\wedge\), \(\vee\), \(\rightarrow\), \(\leftrightarrow\): Binary operators requiring two operands.

**Operands** are the variables or constants upon which the operators act. They can be:

• Literal values like true or false.
• Variables like \(p\), \(q\), \(r\).

In an expression tree, operators are typically represented as internal nodes, whereas operands are leaf nodes, showing the final point of evaluation in the tree structure.

Tree Structure Visualization

An expression tree is a visual representation of a logical formula, breaking down the formula into a structured hierarchy of operations. This visualization helps understand and evaluate formulas in a systematic way.

**How to Construct an Expression Tree:**
1. **Identify the Main Operator:** Begin with the primary logical operator of the formula as the root node.

2. **Break Down Subexpressions:** Each subexpression beneath a binary operator should have its own subtree. Place the operator at the root of the subtree with its operands as children.

3. **Handle Unary Operators:** With unary operators such as negation (\(eg\)), the operator is a parent node, and its operand becomes a leaf node immediately below.

4. **Visual Hierarchy:** Constructing the tree provides a clear hierarchical structure, with levels indicating the operations' precedence and association, making it easier to interpret complex logical statements.

By visualizing logical formulas through trees, each operator and operand's relationships become clear, aiding in better understanding and simplifying the evaluation of the formula.