Problem 57 - Discrete Mathematics with Applications

Question

Construct a truth table for each proposition. $$(p \vee q) \leftrightarrow(p \wedge q)$$

Step-by-Step Solution

Answer

The truth table for the proposition \((p \vee q) \leftrightarrow(p \wedge q)\): | p | q | p ∨ q | p ∧ q | (p ∨ q) ↔ (p ∧ q) | |-----|-----|-------|-------|-------------------| | T | T | T | T | T | | T | F | T | F | F | | F | T | T | F | F | | F | F | F | F | T |

Step 1: Identify possible combinations of truth values for p and q

Start by listing all the possible combinations of truth values for p and q. There are two possible truth values for each variable - True (T) or False (F). For two variables (p and q), we have 2 x 2 = 4 possible combinations.

Step 2: Find the truth values for the disjunction (p ∨ q)

Calculate the truth values for the disjunction \((p \vee q)\) for each combination of truth values for p and q. The disjunction of two propositions is true if at least one of the propositions is true, and false otherwise.

Step 3: Find the truth values for the conjunction (p ∧ q)

Calculate the truth values for the conjunction \((p \wedge q)\) for each combination of truth values for p and q. The conjunction of two propositions is true if both propositions are true, and false otherwise.

Step 4: Find the truth values for the biconditional (p ↔ q)

Calculate the truth values for the biconditional \((p \leftrightarrow q)\) for each combination of truth values for p, q, their disjunction and their conjunction. The biconditional of two propositions is true if both propositions have the same truth value (i.e., either both are true or both are false), and false otherwise.

Step 5: Create the truth table

Construct a truth table with columns for p, q, their disjunction \((p \vee q)\), their conjunction \((p \wedge q)\), and their biconditional \((p \leftrightarrow q)\). Fill in the table with their corresponding truth values from Steps 2, 3, and 4. The truth table for the proposition \((p \vee q) \leftrightarrow(p \wedge q)\): | p | q | p ∨ q | p ∧ q | (p ∨ q) ↔ (p ∧ q) | |-----|-----|-------|-------|-------------------| | T | T | T | T | T | | T | F | T | F | F | | F | T | T | F | F | | F | F | F | F | T |

Key Concepts

Logical Disjunction

Logical disjunction is a fundamental operation in propositional logic, represented by the symbol \( \vee \). This operation is similar to the word "or" in everyday language. When using logical disjunction, the overall statement is true if at least one of the propositions involved is true.

In mathematical terms:

• \( p \vee q \) is true if either \( p \) is true, \( q \) is true, or both are true.
• It is only false if both \( p \) and \( q \) are false.

When looking at a truth table, you would notice the distinct pattern where the result remains true as long as at least one component is true. In our truth table for \( p \vee q \), the result is true in the first three cases, since either \( p \), \( q \), or both are true.

Logical Conjunction

The logical conjunction is another core operation in propositional logic, symbolized by \( \wedge \). It corresponds to the word "and" in language, meaning both conditions must be true for the overall statement to be true.

Specifically:

• \( p \wedge q \) is true only when both \( p \) and \( q \) are true.
• It is false if either \( p \) or \( q \) (or both) are false.

Recall the truth table: \( p \wedge q \) only evaluates to true in the first row, where both \( p \) and \( q \) are true. This highlights how conjunction demands both propositions to hold true for the result to match.

Biconditional

The biconditional operation is symbolized by \( \leftrightarrow \), representing a specific relationship between propositions: it states that two propositions are equivalent in their truth values.

How biconditional works:

• \( p \leftrightarrow q \) is true when both \( p \) and \( q \) are either true or both are false.
• If one is true and the other false, the biconditional is false.

In our example, \((p \vee q) \leftrightarrow (p \wedge q)\) explores the equivalence between the disjunction and conjunction results. The biconditional statement evaluates to true in the first and last rows because those sets of truth values match between \( p \vee q \) and \( p \wedge q \). This illustrates when two logical expressions can be considered equivalent.

Propositional Logic

Propositional logic is a branch of logic dealing with propositions, which are simple, declarative sentences that are either true or false. This type of logic uses symbols to represent logical statements and relationships between them.

Key aspects:

• Fundamental operations include disjunction (\( \vee \)), conjunction (\( \wedge \)), and biconditional (\( \leftrightarrow \)).
• The primary components of propositional logic are propositions, which can be simple or complex.

Consider the exercise:

• "\( p \)" and "\( q \)" are individual propositions.
• Operations like \( \vee \), \( \wedge \), and \( \leftrightarrow \) form new, more complex propositions from the simpler ones.

Truth tables, as shown, provide a structured way to explore all possibilities these expressions can take, illustrating the foundational role propositional logic plays in evaluating logical relationships.