Problem 12 - Discrete Mathematics with Applications

Question

Verify each, where \(f\) denotes a contradiction. $$p \rightarrow(q \vee r) \equiv(p \wedge \sim q) \rightarrow r$$

Step-by-Step Solution

Answer

The equivalence \(p \rightarrow (q \vee r) \equiv (p \wedge \sim q) \rightarrow r\) is verified using a truth table. The truth values of both expressions are the same for all possible values of p, q, and r, confirming their equivalence.

Step 1: Truth Table: Variables and Expressions

Create a column for each variable (p, q, and r) and a column for both expressions, \(p \rightarrow (q \vee r)\) and \((p \wedge \sim q) \rightarrow r\). Step 2: Complete the truth table

Step 2: Truth Table: Values

Fill in the truth values for p, q, and r, and then use these values to determine the truth values for both expressions. Step 3: Compare the truth values of both expressions

Step 3: Comparing Expression Values

If the truth values of both expressions are the same for all possible values of p, q, and r, then the given equivalence holds true. Here's the completed truth table: | p | q | r | \(p \rightarrow (q \vee r)\) | \((p \wedge \sim q) \rightarrow r\) | |-----|-----|-----|--------------------------|-------------------------------| | T | T | T | T | T | | T | T | F | T | F | | T | F | T | T | T | | T | F | F | F | F | | F | T | T | T | T | | F | T | F | T | T | | F | F | T | T | T | | F | F | F | T | T | Since the truth values of both expressions are the same for all possible values of p, q, and r, the equivalence \(p \rightarrow (q \vee r) \equiv (p \wedge \sim q) \rightarrow r\) is verified.

Key Concepts

Truth Table

A truth table is a valuable tool used in propositional logic to determine the truth values of various logical expressions based on all possible combinations of truth values of their components. In our example, we are dealing with the logical expressions \(p \rightarrow (q \vee r)\) and \((p \wedge \sim q) \rightarrow r\).

To construct a truth table, we start by identifying all variables involved, which in this case are \(p\), \(q\), and \(r\). Each can either be true (T) or false (F). Therefore, you get eight possible combinations since each variable has 2 states:

• \(TTT\)
• \(TTF\)
• \(TFT\)
• \(TFF\)
• \(FTT\)
• \(FTF\)
• \(FFT\)
• \(FFF\)

For each combination, you compute the truth value of each expression, allowing you to compare their results easily.

Using a truth table can help us visually compare expressions, making it easier to spot logical equivalences between statements.

Propositional Logic

Propositional logic, also known as propositional calculus or statement logic, is the branch of logic that deals with propositions or statements and their connectives. It's fundamental for understanding logical expressions and how they relate to each other.

In propositional logic, each proposition is a declarative statement that is either true or false. Logical connectives include:

• Negation (\(\sim\)): Inverts the truth value.
• Conjunction (\(\wedge\)): True when both propositions it connects are true.
• Disjunction (\(\vee\)): True if at least one of the propositions it connects is true.
• Implication (\(\rightarrow\)): False only if the first proposition is true and the second is false.

These connective rules allow us to build complex logical expressions by combining simpler ones.
Understanding propositional logic is crucial for verifying logical equivalences, such as the one in our example, where we tested if two statements are consistently of the same truth value, given all variable combinations.

Logical Expressions

Logical expressions are combinations of variables and logical connectives that form a statement that can be either true or false. In the context of our exercise, we see two logical expressions: \(p \rightarrow (q \vee r)\) and \((p \wedge \sim q) \rightarrow r\).

To evaluate these expressions, you need to:

• Interpret the operations correctly.
• Substitute possible truth values of the variables \(p, q,\) and \(r\).
• Determine the truth value of the expressions based on the truth table outcomes.

For instance, the expression \(p \rightarrow (q \vee r)\) means "if \(p\) is true, then \(q\) or \(r\) must be true." Similarly, \((p \wedge \sim q) \rightarrow r\) translates to "if \(p\) is true and \(q\) is false, then \(r\) must be true."

Working with logical expressions allows us to formulate and solve logical problems efficiently, helping to verify equivalences or deduce new information based on given statements.