Problem 46 - Discrete Mathematics with Applications

Question

Let \(t\) be a tautology and \(p\) an arbitrary proposition. Find the truth value of each. $$(\sim t) \rightarrow p$$

Step-by-Step Solution

Answer

In both cases, the expression \((\sim t) \rightarrow p\) evaluates to \(\text{True}\). Therefore, regardless of the truth value of the arbitrary proposition \(p\), the expression is always true.

Step 1: Simplify the expression

Since \(t\) is a tautology, it is always true. Therefore, the negation of a tautology, \(\sim t\), will always be false. Now the expression becomes: \[\text{False} \rightarrow p\]

Step 2: Analyze possible truth values for remaining variables

At this point, we only have one variable left whose value still needs to be determined in order to find the truth value of the expression. Recall that \(p\) is an arbitrary proposition. There are two cases to consider: 1. \(p\) is true 2. \(p\) is false Let's analyze how these cases affect the truth value of the expression.

Step 3: Check the truth value for each case

The expression uses the "implies" operation, which we denote by \(\rightarrow\). Recall the truth table for this operation: \[ \begin{array}{c|c|c} \mathrm{A} & \mathrm{B} & \mathrm{A} \rightarrow \mathrm{B} \\ \hline \mathrm{True} & \mathrm{True} & \mathrm{True} \\ \mathrm{True} & \mathrm{False} & \mathrm{False} \\ \mathrm{False} & \mathrm{True} & \mathrm{True} \\ \mathrm{False} & \mathrm{False} & \mathrm{True} \end{array} \] a) If \(p\) is true, we have: \[\text{False} \rightarrow \text{True}\] According to the truth table for the "implies" operation, this evaluates to \(\text{True}\). b) If \(p\) is false, we have: \[\text{False} \rightarrow \text{False}\] Again, the truth table for the "implies" operation tells us that this evaluates to \(\text{True}\).

Step 4: Conclusion

In both cases, the expression \((\sim t) \rightarrow p\) evaluates to \(\text{True}\). Therefore, regardless of the truth value of the arbitrary proposition \(p\), the expression is always true.

Key Concepts

Truth Tables

A truth table is a very useful tool in logic and mathematics. It helps us understand how logical expressions and operations behave under different conditions. Essentially, a truth table lists all possible combinations of truth values for a set of propositions. By evaluating each possible scenario, we can determine the outcome of a complex logical expression.

When constructing a truth table, you list all variables involved and consider every possible combination of true (T) and false (F) values each variable might have. For binary operations, there are often four possible scenarios: both values true, one true and one false, and both false. We then look at what happens in each scenario to understand the logical relationship expressed in the operation.

In our exercise, the truth table for the implication (denoted by \(\rightarrow\)) shows us that unless the first part (the antecedent) is true and the second part (the consequent) is false, the whole implication statement remains true. So, by using a truth table, we quickly found that regardless of the truth value of proposition \(p\), the original expression was always true.

Propositional Logic

Propositional logic is a branch of logic that studies the ways in which propositions (statements that can be either true or false) can be combined and manipulated to form new propositions. Unlike other forms of logic, propositional logic doesn't require us to understand the content of the propositions, only their truth values.

In propositional logic, the basic building blocks are propositions, often represented by letters such as \(p\) and \(q\). These propositions can be combined using logical connectives like AND (\(\land\)), OR (\(\lor\)), NOT (\(\sim\)), and IMPLIES (\(\rightarrow\)). The resulting expressions reveal new truths from the original propositions.

Our exercise involved an arbitrary proposition \(p\) and a tautology \(t\). By understanding that a tautology is always true, we simplified the original expression. The expression then became a focus on how \(p\) interacts with the negation of \(t\) via the implies operation in logical terms.

Implication Operation

The implication operation, often written as \(\rightarrow\), is a fundamental aspect of logical reasoning. It connects two propositions in such a way that the entire expression is false only if the first proposition (the antecedent) is true while the second (the consequent) is false.

Understanding the implication operation requires acquaintance with its truth table, which helps clarify why \(A\rightarrow B\) is mostly true. Here's a brief breakdown:

• If \(A\) is true and \(B\) is true, then \(A \rightarrow B\) is true.
• If \(A\) is true and \(B\) is false, then \(A \rightarrow B\) is false.
• If \(A\) is false and \(B\) is true, then \(A \rightarrow B\) is true.
• If \(A\) is false and \(B\) is false, then \(A \rightarrow B\) is true.

In our example, \(\sim t\rightarrow p\), the operation played a crucial role. Since the antecedent (\(\sim t\)) is always false, the truth table assured us that the entire expression evaluates to true regardless of whether \(p\) is true or false. This is why we found that no matter the proposition \(p\), it results in a true statement.