Problem 29 - Discrete Mathematics with Applications

Question

Give the truth value of each proposition, using the given information. \(p \vee q,\) where \(\sim p\) is false.

Step-by-Step Solution

Answer

The truth value of the given proposition \(p \vee q\) is true, because \(p\) is true.

Step 1: Determine the truth value of p

Since the negation of \(p\), or \(\sim p\), is given to be false, we can conclude that the truth value of \(p\) must be true. In other words: If \(\sim p\) is false, then \(p\) is true.

Step 2: Analyze the disjunction symbol

The disjunction symbol \(p \vee q\) represents the logical "OR" operation. This means that the proposition \(p \vee q\) will be true if at least one of the propositions \(p\) or \(q\) is true. In other words, \(p \vee q\) is true if \(p\) is true or \(q\) is true or both are true.

Step 3: Determine the truth value of the given proposition

Now we know the truth value of \(p\) and the definition of the disjunction symbol, we can determine the truth value of the given proposition \(p \vee q\). Since \(p\) is true (from Step 1), and knowing the definition of the disjunction symbol from Step 2, we can conclude that: \(p \vee q\) is true, because \(p\) is true.

Key Concepts

Truth Value

The truth value of a proposition in logic tells us whether the given proposition is true or false.
In propositional logic, propositions are statements that can concretely have this binary property - true or false.

• If a proposition is true, we say it has a truth value of "true" or simply "T".
• If a proposition is false, it has a truth value of "false" or "F".

The truth value is very much like a yes or no answer, confirming the status of a given proposition. For instance, a statement like "The sky is blue" would be true, while "The sky is pink" might generally be false under normal daylight conditions. Remember, in propositional logic, these truth values help us evaluate logical sentences and expressions. In the exercise at hand, determining the truth value of each part assists in solving the overall problem.

Logical Disjunction

The logical disjunction is a fundamental concept in propositional logic, and it is represented by the symbol \(\vee\).
Logically, it operates as the "OR" function in everyday language.

• The statement \(p \vee q\) is true if either proposition \(p\) is true, or proposition \(q\) is true, or both.
• Only if both \(p\) and \(q\) are false does the disjunction \(p \vee q\) become false.

For example, if you were told "We can have pizza or pasta for dinner," and both options are available, then the statement is true if you choose at least one option. Nothing blocks you from having both, but you only need one to fulfill the condition. In propositional logic, the power of disjunction allows us to combine statements in such a way that increases our chances of arriving at a true proposition. Applying this concept, if \(p\) is true (based on our determination of truth value), the disjunction \(p \vee q\) must also be true, confirming the outcome of our exercise.

Negation

Negation in propositional logic is a way of making a proposition express the exact opposite of its original meaning.
It is denoted by \(\sim\) (not, negation) placed before a proposition.

• Negating a true proposition yields a false one.
• Conversely, negating a false proposition makes it true.

To illustrate, consider you have the proposition "It is sunny." Negating it would result in "It is not sunny." If "It is sunny" is true, then negating it confirms "It is not sunny" as false. In the exercise, we started with the information that \(\sim p\) is false. This directly led us to conclude that \(p\) must be true, because negating a proposition flips its truth value. Understanding negation is crucial when interpreting not only single propositions but also when dealing with complex logical expressions where multiple layers of negations might exist.