Problem 3 - Discrete Mathematics with Applications

Question

Verify each, where \(f\) denotes a contradiction. $$\sim(\sim p) \equiv p$$

Step-by-Step Solution

Answer

The given logical equivalence \(\sim(\sim p) \equiv p\) is verified by creating truth tables for both \(\sim(\sim p)\) and \(p\), evaluating the truth values for each possible value of \(p\), and comparing the resulting tables. The truth values for \(\sim(\sim p)\) and \(p\) are found to be the same for each possible value of \(p\), confirming that the logical equivalence holds true.

Step 1: Create a truth table for \(\sim(\sim p)\)

We'll create a truth table with two columns: one for \(p\) representing the possible truth values, and another for \(\sim(\sim p)\), which represents the given logical expression. | \(p\) | \(\sim(\sim p)\) | |-------|-----------------| | T | | | F | |

Step 2: Evaluate \(\sim(\sim p)\) for each value of \(p\)

We'll compute the value of the given expression \(\sim(\sim p)\) for each possible value of \(p\), considering that \(\sim\) is the negation (NOT) logical operator, which reverses the truth value of the inner statement. | \(p\) | \(\sim(\sim p)\) | |-------|-----------------| | T | T | | F | F |

Step 3: Create a truth table for \(p\)

Create a truth table with the possible truth values of \(p\). | \(p\) | |-------| | T | | F |

Step 4: Compare the truth tables

Now, we'll compare the truth tables of \(\sim(\sim p)\) and \(p\). If they have the same truth values, the logical equivalence holds true. | \(p\) | \(\sim(\sim p)\) | \(p\) | |-------|-----------------|-------| | T | T | T | | F | F | F | From the comparison, we can see that the truth values of \(\sim(\sim p)\) and \(p\) are the same for each possible value of \(p\). Therefore, the given logical equivalence is verified: \[\sim(\sim p) \equiv p.\]

Key Concepts

Truth Table

A truth table is an essential tool in logic, used to determine the truth value of a logical expression based on all possible combinations of truth values of its components. It works by laying out all potential scenarios for each variable within the logical statement. By doing this, we can systematically evaluate the truth or falsity of the statement for every possible situation.

To construct a truth table, follow these basic steps:

• Identify all the variables within your logical statement.
• List all possible truth values for these variables. For a single variable, there are two values: true (T) and false (F).
• Calculate the truth value of the expression for each combination of assigned truth values of variables.

The truth table helps in visually understanding how different combinations affect the overall expression. This method is precise and leaves no room for error, which is crucial in verifying logical equivalences or proving contradictions.

Logical Equivalence

Logical equivalence is a crucial concept in logic and reasoning. Two statements are logically equivalent if they have identical truth values in all possible scenarios. Essentially, both statements say the same thing in terms of their truth.When working with logical equivalence, you need primarily to:

• Compare truth tables for both expressions to ensure that their truth values match for every possible situation.
• Identify rules and laws, such as De Morgan's Laws or Double Negation, that can simplify logical statements.

The importance of logical equivalence lies in simplifying complex logical statements. For example, knowing that \(eg(eg p) \equiv p\) means you can substitute one for the other in a proof, ensuring simplicity without losing correctness. Understanding how to identify and use logical equivalences is vital in mathematical proofs, computer algorithms, and artificial intelligence, where simplification can reduce computation time.

Negation

Negation is a fundamental operation in logic, usually represented by the symbol "\(\sim\)" or sometimes "NOT." This operation flips the truth value of a given statement. If a statement is true, its negation is false, and vice versa.

To understand negation, consider:

• If "p" is "The sky is blue," then "\(\sim p\)" (negation of p) would be "The sky is not blue."
• If "p" is true, "\(\sim p\)" is false.

Negation is particularly useful when creating expressions that need to represent "not" conditions. Furthermore, in logical proofs, double negation is quite significant, such that negating a negation returns the original truth value: \(\sim(\sim p) \equiv p\). This principle is often used to simplify logical expressions and is essential to establish logical equivalences.