Problem 4 - Discrete Mathematics with Applications

Question

Verify each, where \(f\) denotes a contradiction. (See Table \(1.14 . )\) $$ p \wedge p \equiv p $$

Step-by-Step Solution

Answer

To verify the equivalence \(p \wedge p \equiv p\), we can use a truth table: | p | p ∧ p | |---|-------| | T | T | | F | F | Since the truth values of statement \(p\) are identical to the truth values of the expression \(p \wedge p\), the given equivalence holds true: $$ p \wedge p \equiv p $$

Step 1: Set up the truth table

First, set up a truth table with one statement, \(p\), and the resulting truth value of the given expression, \(p \wedge p\). | p | p ∧ p | |---|-------| | T | | | F | |

Step 2: Determine the truth values

Next, determine the truth values for the given expression, \(p \wedge p\), based on the truth values of statement \(p\). | p | p ∧ p | |---|-------| | T | | | F | | Since 'and' operation \(\wedge\) means that the result is true when both inputs are true, apply this definition to the table: | p | p ∧ p | |---|-------| | T | T | | F | F |

Step 3: Compare the truth values

Finally, compare the truth values of statement \(p\) and the expression \(p \wedge p\): | p | p ∧ p | |---|-------| | T | T | | F | F | The truth values of statement \(p\) are identical to the truth values of the expression \(p \wedge p\). Therefore, the given equivalence holds true: $$ p \wedge p \equiv p $$

Key Concepts

Contradiction

Understanding the concept of contradiction in logic is essential for students navigating through complex logical premises. A contradiction occurs when a statement is logically unsound, meaning it is false under all conditions. For instance, the statement 'It is raining and it is not raining' is a contradiction because both conditions cannot be true simultaneously. In truth tables, a contradiction is represented by a statement or compound statement that yields a false value for every possible interpretation of its components. This is an essential concept when working with logical expressions, as identifying contradictions helps in simplifying arguments and proving the invalidity of certain statements.

Logical Conjunction

The logical conjunction, denoted by the symbol \(\wedge\), plays a pivotal role in logical operations. This operation, commonly known as 'AND', combines two or more statements and is true if and only if all the individual statements are true. In the context of our original exercise, \(p \wedge p\) equates to \(p\) because the conjunction of a statement with itself will always yield the original statement's truth value.

Using a truth table, we depict all possible truth values of the involved statements and then apply the definition of \(\wedge\) to find the result. This method clarifies how conjunctions operate within logical expressions, reinforcing the understanding of how complex statements can be constructed and evaluated.

Equivalence Relation

In logic theory, an equivalence relation is a connection between two statements or propositions indicating they have the same truth value. The symbol for logical equivalence is \(\equiv\), and it demonstrates that two statements are true under exactly the same conditions or false under the same conditions. The exercise provided, \(p \wedge p \equiv p\), is a straightforward example of an equivalence relation as the compound statement \(p \wedge p\) is always equivalent to \(p\) itself; their truth values match perfectly. Grasping equivalence relations enables students to understand when two logical formulas can replace each other without altering the truth conditions of a logical expression, which is a cornerstone of logical reasoning and proof.