Problem 24 - Discrete Mathematics with Applications

Question

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

Step-by-Step Solution

Answer

The truth table for the proposition \(\sim(\sim p \vee q)\) is: | \(p\) | \(q\) | \(\sim p\) | \(\sim p \vee q\) | \(\sim(\sim p \vee q)\) | |-------|-------|--------|-------------|-------------------| | T | T | F | T | F | | T | F | F | F | T | | F | T | T | T | F | | F | F | T | T | F |

Step 1: Identify the logical operations involved

The proposition has two different logical operations: negation and disjunction. The proposition can also be written as \(\lnot(\lnot p \lor q)\), where \(\lnot\) is negation and \(\lor\) is disjunction. The truth value of the proposition depends on the truth values of \(p\) and \(q\).

Step 2: Determine all possible combinations of truth values

The variables \(p\) and \(q\) can each take on two different truth values, true (T) or false (F), which means there are four possible combinations for the proposition: 1. \(p\) is T and \(q\) is T 2. \(p\) is T and \(q\) is F 3. \(p\) is F and \(q\) is T 4. \(p\) is F and \(q\) is F

Step 3: Calculate the truth values of the inner parts

First, we need to evaluate the truth values of \(\lnot p\) and \(\lnot p \lor q\) for each combination of values for \(p\) and \(q\): 1. If \(p\) is T and \(q\) is T, then \(\lnot p\) is F and \(\lnot p \lor q\) is T 2. If \(p\) is T and \(q\) is F, then \(\lnot p\) is F and \(\lnot p \lor q\) is F 3. If \(p\) is F and \(q\) is T, then \(\lnot p\) is T and \(\lnot p \lor q\) is T 4. If \(p\) is F and \(q\) is F, then \(\lnot p\) is T and \(\lnot p \lor q\) is T

Step 4: Calculate the truth values of the whole proposition

Now, we can determine the truth value of the whole proposition, \(\lnot(\lnot p \lor q)\): 1. If \(\lnot p \lor q\) is T, then \(\lnot(\lnot p \lor q)\) is F 2. If \(\lnot p \lor q\) is F, then \(\lnot(\lnot p \lor q)\) is T 3. If \(\lnot p \lor q\) is T, then \(\lnot(\lnot p \lor q)\) is F 4. If \(\lnot p \lor q\) is T, then \(\lnot(\lnot p \lor q)\) is F

Step 5: Construct the truth table

Finally, we can create the truth table for the proposition \(\lnot(\lnot p \lor q)\): | \(p\) | \(q\) | \(\lnot p\) | \(\lnot p \lor q\) | \(\lnot(\lnot p \lor q)\) | |-------|-------|--------|-------------|-------------------| | T | T | F | T | F | | T | F | F | F | T | | F | T | T | T | F | | F | F | T | T | F |

Key Concepts

Logical Operations

Logical operations are the building blocks of most logical expressions. They allow us to take basic propositions and combine them to form complex statements. Just like arithmetic operations (add, subtract, multiply, divide) are used to manipulate numbers, logical operations manipulate truth values: true (T) and false (F). In the context of this exercise, we are working with two fundamental logical operations: negation and disjunction.

Understanding how these operations work will enable you to construct and evaluate more involved logical propositions. Practicing these operations with truth tables is an effective way to become familiar with how different combinations of propositions yield various truth values.

Negation

Negation is a fundamental logical operation that flips the truth value of a proposition. If a statement is true, its negation is false, and if a statement is false, its negation is true. The symbol used for negation is \lnot, which can be read as "not." You can think of it as a logical "not" that precedes a proposition and inverses its truth value.

For example, given a proposition \(p\):

• If \(p\) is true, then \(\lnot p\) is false.
• If \(p\) is false, then \(\lnot p\) is true.

In this exercise, negation is applied twice: once directly to \(p\) to form \(\lnot p\), and then again to the result of the disjunction operation \(\lnot p \lor q\). This shows how negation can be used in nested operations to produce a complex proposition.

Disjunction

Disjunction is another crucial logical operation that combines two propositions. It is denoted by the symbol \(\lor\) and reads as "or." In logical terms, a disjunction is true if at least one of the propositions is true. This operation means that if either of the component propositions is true, or both are true, the resulting compound statement is true.

In the context of the given exercise, the disjunction operation \(\lnot p \lor q\):

• Is true if \(\lnot p\) is true or \(q\) is true.
• Is false only if both \(\lnot p\) and \(q\) are false.

Analyzing how disjunction works within a truth table helps us see how logical OR operates across different combinations of truth values for the propositions involved.

Propositions

Propositions are the basic statements in logic that declare a fact or an opinion, which can either be true or false. In this exercise, \(p\) and \(q\) are the propositions that form the basis for constructing the truth table. Each of these propositions can independently be true or false, leading to various combinations.

The power of propositions lies in their ability to be combined using logical operations like negation and disjunction, transforming into more complex expressions. The evaluation of these expressions through truth tables enables us to interpret their overall truth values for all possible scenarios. By mastering how propositions and their combinations work, you have a strong foundation to approach complex logical problems and deduce their outcomes effectively.