Problem 9 - Thinking Mathematically

Question

Use a truth table to determine whether the two statements are equivalent. \((p \wedge q) \vee r, p \wedge(q \vee r)\)

Step-by-Step Solution

Answer

The given expressions \((p \land q) \lor r\) and \(p \land (q \lor r)\) are not equivalent.

Step 1: List all Variable Combinations

List each variable combination for \(p\), \(q\), and \(r\). Since each variable can take the boolean values True or False, with three variables there are \(2^3 = 8\) different possible combinations.

Step 2: Compute Value for First Expression

For each variable combination compute the value of the first expression \((p \land q) \lor r\). Use the logic operation AND (\(\land\)) and OR (\(\lor\)) rules: AND is true if both premises are true, else is false; OR is true if at least one of the premises is true.

Step 3: Compute Value for Second Expression

For each variable combination compute the value of the second expression \(p \land (q \lor r)\). Use the logic operation AND (\(\land\)) and OR (\(\lor\)) rules as before.

Step 4: Compare the Results

Compare the resulting truth values of both expressions. If they are same for all combinations of variable truth values, then the two expressions are equivalent. If not, then they are not equivalent.

Key Concepts

Propositional Logic

Propositional logic is a branch of logic that deals with statements or 'propositions' that are either true or false. Each proposition is represented by a variable, such as 'p', 'q', or 'r', which can take on the value of true or false. These propositions can be combined using logical operators to create more complex expressions. The primary logical operators in propositional logic are 'NOT' (eg), 'AND' (), 'OR' (), cond 'OR' (), and 'OR' ().

Logical Conjunction

Logical conjunction, commonly represented by the symbol , refers to the logical operation 'AND'. This operator combines two propositions and returns true only if both propositions are true. If either of the propositions is false, or if both are false, the result of a conjunction is false. For example, for the propositions 'p' and 'q', the conjunction 'p ' 'q' will only be true when both 'p' and 'q' are true.

Imagine you have two statements: 'It is raining' (p) and 'I have an umbrella' (q). The conjunction of these two statements 'It is raining AND I have an umbrella' (p ) 'q' represents the scenario where both conditions are simultaneously met. If either it's not raining or you don't have an umbrella, then the conjunction is not satisfied and is therefore false.

Logical Disjunction

Logical disjunction is the logical operation 'OR' denoted by the symbol . It combines two propositions and returns true if at least one of the propositions is true. The expression 'p ' 'q' is true if 'p' is true, or 'q' is true, or if both are true. It is only false when both 'p' and 'q' are false.

Using our previous example, if 'p' represents 'It is raining' and 'q' represents 'I have an umbrella', the disjunction 'It is raining OR I have an umbrella' (p ) 'q' would be true if it is raining, or if you have an umbrella, or if both conditions apply. This operation allows flexibility, such that as long as one of the conditions is met, the overall expression yields true.