Question

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

Short answer

(a) OR as root, AND with p and q on left, r on right. (b) Implication as root, nested implication on left, r on right. (c) Implication as root with p, nested implication of q and r on right.

Step-by-step solution

Interpret the Formula (a)

The formula given is \( (p \wedge q) \vee r \). This expression represents a Boolean logic formula composed of logical connectives. Our goal is to break down the formula into its components: literals \(p\), \(q\), and \(r\), logical AND \(\wedge\), and logical OR \(\vee\).

Construct the Expression Tree for (a)

An expression tree is a binary tree where each internal node is an operator, and each leaf node is an operand. For \( (p \wedge q) \vee r \):- The root node is \(\vee\).- The root's left child is another operator \(\wedge\), with children \(p\) and \(q\).- The root's right child is \(r\).

Interpret the Formula (b)

The formula provided is \( (p \rightarrow q) \rightarrow r\). This involves logical implications.- \(\rightarrow\) denotes implication, and we'll build the expression tree similarly to step 2, considering implications as binary operations.

Construct the Expression Tree for (b)

For the formula \( (p \rightarrow q) \rightarrow r \):- The root node is the second implication \(\rightarrow\).- The root's left child is the first implication \(\rightarrow\). - This node's left and right children are \(p\) and \(q\) respectively.- The root's right child is \(r\).

Interpret the Formula (c)

The formula is \( p \rightarrow (q \rightarrow r) \). This indicates nested implications.- Recognize that each \(\rightarrow\) is an operator and \(p\), \(q\), and \(r\) are operands.

Construct the Expression Tree for (c)

For \( p \rightarrow (q \rightarrow r) \):- The root node is an implication \(\rightarrow\).- The root's left child is \(p\).- The root's right child is another implication \(\rightarrow\). - This second implication has left child \(q\) and right child \(r\).

Key concepts

Boolean logic

Boolean logic is a branch of mathematics dealing with variables that have two distinct values: true or false. It is foundational for digital circuit design, programming, and formulating logical expressions. Boolean logic operates on principles of binary computation, where logical statements can be expressed as true or false. This simple logic forms the backbone of computer science and helps solve complex expressions in a structured way.

In Boolean logic, we primarily utilize variables, operators, and logical connectives to form expressions. Variables, like the literals in our exercise (\(p, q,\) and \(r\)), represent truth values, whereas operators manipulate these values to achieve desired outcomes. The fundamental operators in Boolean logic include AND (\(\land\)), OR (\(\lor\)), and NOT (\(eg\)). These operators help in combining or modifying true or false values. For instance, the expression \(p \land q\) is true only if both \(p\) and \(q\) are true.

Understanding Boolean logic assists in decomposing logic problems into manageable parts. This is essential while constructing expression trees, as they help visualize and process logical statements using a hierarchical structure.

Logical connectives

Logical connectives are the symbols or words used in logic to connect two or more statements or propositions. These connectives help form complex logical expressions and dictate how combined individual truth values result in a single outcome.

The main logical connectives are:

• AND (\(\land\)): Outputs true if both operands are true.
• OR (\(\lor\)): Outputs true if at least one operand is true.
• NOT (\(eg\)): Reverses the truth value of a single operand.
• IMPLIES (\(\rightarrow\)): Outputs false only if the first operand is true and the second is false.
• BICONDITIONAL (\(\leftrightarrow\)): True if both operands have the same truth value.

Logical connectives are crucial in defining the relationships within logical structures, such as expression trees. These trees use connectives as nodes to demonstrate the sequence and hierarchy of operations. For example, in the expression tree for \((p \land q) \lor r\), \(\lor\) is the root connective, guiding how the logical flow leads to the ultimate truth value.

Grasping logical connectives allows for the dissection and interpretation of complex logical formulas and helps in computer programming logic, artificial intelligence, and mathematical proofs.

Logical implications

Logical implications are a type of logical connective that express a conditional relationship between statements. These implications are essentially 'if-then' scenarios, where the truth of one statement guarantees the truth of a subsequent statement.

For a statement \(p \rightarrow q\), it is only false when \(p\) is true and \(q\) is false. Otherwise, it is considered true. This uniqueness makes implications particularly useful in reasoning and inference, where conditional rules are applied.

Expression trees featuring logical implications clearly illustrate the sequence of conditional operations. For instance, in an expression tree for \((p \rightarrow q) \rightarrow r\), each node reflects an implication, showcasing how the operation unfolds. The implication connective (\(\rightarrow\)) as the top node guides the direction of evaluating truth values, reflecting the hierarchical nature of implications.

By understanding logical implications, you can analyze and evaluate scenarios effectively, using them as aids in proofs, algorithms, and logical decision-making processes. Emphasizing the conditionals helps students comprehend the flow of logic in their reasoning, critical in fields such as mathematics, computer science, and philosophical arguments.