Problem 73 - Discrete Mathematics with Applications

Question

Let \(p, q,\) and \(r\) be simple propositions with \(t(p)=1, t(q)=0.3,\) and \(t(r)=\) \(0.5 .\) Compute the truth value of each, where \(s^{\prime}\) denotes the negation of the statement \(s\) $$(p \vee q)^{\prime}$$

Step-by-Step Solution

Answer

The truth value of the given expression \((p \vee q)'\) is 0. To compute this, we first found the disjunction (p ∨ q) to be true, with a truth value of 1. Then, we computed the negation by subtracting the truth value of (p ∨ q) from 1, resulting in a truth value of 0 for \((p \vee q)'\).

Step 1: Evaluate p ∨ q

To evaluate the expression (p ∨ q), we will use the truth values of p and q, which are provided. In this case, p is true (t(p) = 1) and q is 0.3 (t(q) = 0.3). The disjunction (p ∨ q) is true if either p or q or both are true. Since p is true, the disjunction (p ∨ q) is true. Let's denote the truth value of (p ∨ q) as t(p ∨ q) = 1.

Step 2: Negate (p ∨ q)

Now we are asked to compute the negation of (p ∨ q), denoted as (p ∨ q)'. To find the negation of a statement, we subtract its truth value from 1. In this case, since the truth value of (p ∨ q) is 1, we have: \((p \vee q)' = 1 - t(p \vee q) = 1 - 1 = 0 \)

Step 3: Conclusion

The truth value of the given expression \((p \vee q)'\) is 0.

Key Concepts

Truth Value

In propositional logic, a truth value is used to determine whether a proposition is true or false.

• If a truth value is 1, it means the proposition is true.
• If it is 0, the proposition is false.

There are also fuzzy truth values that can exist in a range between 0 and 1. This allows a proposition to be partially true. In the given exercise, the truth values for propositions are: - \(t(p) = 1\), meaning proposition \(p\) is completely true. - \(t(q) = 0.3\), indicating proposition \(q\) is partially true.- And \(t(r) = 0.5\), showing that \(r\) has truth value that is exactly halfway between completely true and completely false.Understanding these values helps in computing expressions and their outcomes when combined with logical operators, such as negation and disjunction.

Negation in Logic

Negation in logic essentially reverses the truth value of a proposition. It is depicted as the prime symbol \((s')\) when applied to a statement \(s\). The truth value of a negated statement can be found by subtracting the truth value of the original statement from 1. For example, if the truth value of a statement \(s\) is 0.7, the negation of \(s\), written as \(s'\), would have a truth value of:\[s' = 1 - 0.7 = 0.3\]In the exercise, the negation \((p \vee q)'\) was found by:- First determining the truth value of \(p \vee q\). - Then subtracting that value from 1.Negation is a fundamental operation in logic that helps switch between true and false as needed within logical expressions.

Disjunction in Logic

Disjunction in logic uses the \(\vee\) symbol and corresponds to the word "or." In the expression \(p \vee q\), the result is true if either \(p\) or \(q\)—or both—are true.This logic rule makes disjunction a very inclusive operation since it only requires at least one true component to result in true. In the exercise:

• \(p\) with \(t(p) = 1\) is true.
• \(q\) with \(t(q) = 0.3\) is partially true but not false.

Since \(p\) is completely true, the disjunction \(p \vee q\) yields a truth value of 1.Understanding disjunction is crucial because it helps to evaluate how multiple propositions combine in logical statements, leading to determining if the overall expression is true.