Question

Write the truth tables for the following formulas, Use the truth table to determine whether any of these formulas is a tautology. (a) \(((p \rightarrow q) \wedge(q \rightarrow r)) \rightarrow(p \leftrightarrow r)\) (b) \(((p \rightarrow q) \wedge(q \rightarrow r)) \rightarrow(p \rightarrow r)\) (c) \(((p \rightarrow q) \rightarrow r) \rightarrow(p \rightarrow(q \rightarrow r))\) (d) \((p \rightarrow(r \vee q)) \rightarrow((p \rightarrow r) \vee(p \rightarrow q))\) (e) \((p \rightarrow(r \wedge q)) \rightarrow((p \rightarrow r) \vee(p \rightarrow q))\) (f) \(((p \rightarrow q) \rightarrow q) \rightarrow p\)

Short answer

Formulas (b) and (d) are tautologies.

Step-by-step solution

Set up the truth table structure

To analyze the formulas, we must first set up a truth table for each formula. For each statement, list all possible truth values for the variables involved (p, q, r). There are 8 possible combinations of truth values (True or False) for p, q, and r.

Evaluate Formula (a)

1. List columns for each component: \(p\), \(q\), \(r\), \(p \rightarrow q\), \(q \rightarrow r\), \((p \rightarrow q) \wedge (q \rightarrow r)\), and \((p \leftrightarrow r)\).2. Calculate the truth values for each component using logical rules.3. Analyze the final component \(((p \rightarrow q) \wedge (q \rightarrow r)) \rightarrow (p \leftrightarrow r)\). If it is always True, the formula is a tautology.

Evaluate Formula (b)

1. List columns for each component: \(((p \rightarrow q) \wedge (q \rightarrow r))\), \((p \rightarrow r)\).2. Calculate the truth values for each sub-expression using logical rules.3. Determine the truth of \(((p \rightarrow q) \wedge (q \rightarrow r)) \rightarrow (p \rightarrow r)\) by verifying in each row if the implication holds. Check if this is a tautology.

Evaluate Formula (c)

1. List columns for each component: \(p\), \(q\), \(r\), \(p \rightarrow q\), \((p \rightarrow q) \rightarrow r\), \(q \rightarrow r\), and \(p \rightarrow(q \rightarrow r)\).2. Calculate the truth values for each setup.3. Evaluate \(((p \rightarrow q) \rightarrow r) \rightarrow (p \rightarrow (q \rightarrow r))\) to check if it always holds true, thus determining if it is a tautology.

Evaluate Formula (d)

1. Set up truth table columns: \((p \rightarrow (r \vee q))\), \((p \rightarrow r)\), \((p \rightarrow q)\), and \(((p \rightarrow r) \vee (p \rightarrow q))\).2. Fill out the table for each component.3. Verify \((p \rightarrow (r \vee q)) \rightarrow ((p \rightarrow r) \vee (p \rightarrow q))\) across all rows to check for tautology.

Evaluate Formula (e)

1. Write truth table columns: \((p \rightarrow (r \wedge q))\), \((p \rightarrow r)\), \((p \rightarrow q)\), and \(((p \rightarrow r) \vee (p \rightarrow q))\).2. Calculate each expression for all truth combinations.3. Conclude if \((p \rightarrow (r \wedge q)) \rightarrow ((p \rightarrow r) \vee (p \rightarrow q))\) is a tautology by checking if it consistently results in true.

Evaluate Formula (f)

1. Construct columns for each part: \(p\), \(q\), \(p \rightarrow q\), \((p \rightarrow q) \rightarrow q\), and \((((p \rightarrow q) \rightarrow q) \rightarrow p)\).2. Use logical operations to fill in columns.3. Analyze the final column of the formula \(((p \rightarrow q) \rightarrow q) \rightarrow p\) to determine if it is always True for any combination (indicating a tautology).

Key concepts

Tautology

A tautology is a fascinating concept in logic. It's like a statement that can't be false no matter what. Imagine you have a formula and you want to check if it's always true. That's where tautology comes in!
- When a logical statement is a tautology, it will be true regardless of the truth values of its components. - It's like a mathematical certainty. For example, the statement "Either it will rain tomorrow, or it will not rain tomorrow" is a tautology because it covers all possibilities.In the context of truth tables, a formula is considered a tautology if every possible row in the table evaluates to true. For example, in a logical formula involving variables like \(p\), \(q\), and \(r\), no matter whether these variables are true or false, the formula will still return "true."
Understanding tautologies is essential because they form the backbone of logical reasoning. They help in verifying the correctness of arguments by assuring that the structure of the logical formula is inherently reliable.

Logical Formulas

Logical formulas are the building blocks of logical reasoning. They are expressions constructed using variables and logical operations like AND, OR, and NOT.
- Variables like \(p\), \(q\), and \(r\) represent propositions which can be true or false. - Logical formulas combine these variables using logical operators to create more complex expressions.The beauty of logical formulas is their versatility and ability to represent intricate ideas in a structured way. You might see them in mathematics and computer science, where expressing scenarios clearly and precisely is crucial.
Understanding logical formulas enables you to use truth tables effectively to determine whether a statement is true, false, or a tautology. By evaluating each component of a formula separately, you can observe how the entire expression behaves under different conditions.

Implication

Implication is a key concept in logic represented usually by the arrow operator \(\rightarrow\). It connects two statements together like "if this, then that."
- In a formula such as \(p \rightarrow q\), \(p\) is called the antecedent, and \(q\) is the consequent.- The implication \(p \rightarrow q\) is true in all cases except when \(p\) is true and \(q\) is false.This might seem counterintuitive at first, but in logic, \(p \rightarrow q\) promises that "if \(p\) happens, \(q\) will also happen." If \(p\) doesn't happen, then it's not breaking this promise.
Through implications, logical expressions can create complex relationships, especially when used in combination with other logical operators. This is why understanding how implication works is crucial for devising correct logical formulas and solving related problems.

Logical Operators

Logical operators are the glue that holds logical formulas together, allowing us to form complex expressions from simpler ones.
- Common logical operators include AND (\(\wedge\)), OR (\(\vee\)), NOT (\(eg\)), and IMPLIES (\(\rightarrow\)). - These operators work similarly to how conjunctions and disjunctions link ideas in human languages.

1. AND (\(\wedge\)): Returns true if both connected statements are true.
2. OR (\(\vee\)): Returns true if at least one of the statements is true.
3. NOT (\(eg\)): Flips the truth value of a statement. If a statement is true, NOT makes it false, and vice versa.
4. IMPLIES (\(\rightarrow\)): Explained further in the "Implication" section, it connects statements in a conditional manner.

By using these operators, we can evaluate complex logical formulas in a truth table, checking for tautologies, contradictions, or contingencies to aid logical reasoning and problem-solving.