Problem 13 - Discrete Mathematics with Applications

Question

Give the simplest possible conclusion in each argument. Assume each premise is true. $$\begin{aligned} &p \leftrightarrow q\\\ &\sim p \vee r\\\ &\sim r \end{aligned}$$

Step-by-Step Solution

Answer

The simplest possible conclusion for this logical argument is \(q\).

Step 1: Rewrite the Biconditional as Two Conditionals

The first premise is a biconditional, \(p \leftrightarrow q\). We can rewrite it as two separate conditional statements, \(p\rightarrow q\) and \(q \rightarrow p\), to simplify the analysis.

Step 2: Apply Modus Tollens to Derive New Information

With the third premise, \(\sim r\), apply Modus Tollens on the second premise (\(\sim p \vee r\)) to derive a new conclusion: $$\sim(\sim p \vee r) \rightarrow r$$ $$\sim r \rightarrow \sim(\sim p \vee r)$$ Since we know that \( \sim r\) is true, we can deduce that \(\sim(\sim p \vee r)\) is also true.

Step 3: Use De Morgan's Law

Now, we can apply De Morgan's Law to the newly derived premise, \(\sim(\sim p \vee r)\): $$\sim(\sim p \vee r) \equiv \sim(\sim p) \wedge \sim r$$ $$\sim(\sim p) \wedge \sim r \equiv p \wedge \sim r$$

Step 4: Combine Information from Steps 1 and 3

We know from step 1 that \(p \rightarrow q\). Combining this with the information derived in step 3, which is \(p \wedge \sim r\), we can use the rule of simplification to deduce that if \(p \wedge \sim r\) is true, then \(p\) must also be true.

Step 5: Apply Modus Ponens to Make a Conclusion

Since we've deduced that \(p\) is true, we can apply Modus Ponens using the conditional statement derived in step 1, \(p \rightarrow q\): $$p \rightarrow q$$ $$p$$ Therefore, the simplest possible conclusion for this logical argument is: $$\boxed{q}$$

Key Concepts

Biconditional

The biconditional is a logical connective used to join two statements into one proposition. When you see a statement like \(p \leftrightarrow q\), it reads as "\(p\) if and only if \(q\)." It means both statements must either be true together or false together.

Biconditionals can be broken down into two separate conditionals for easier analysis:

• \(p \rightarrow q\): If \(p\) is true, then \(q\) is true.
• \(q \rightarrow p\): If \(q\) is true, then \(p\) is true.

This transformation helps in logical reasoning because conditionals are often simpler to work with than biconditionals. Knowing how to split a biconditional into conditionals is a powerful skill when solving logical problems.

Modus Tollens

Modus Tollens is a form of logical reasoning that involves deduction. It is used to infer a conclusion by proving the opposite of the necessary condition to be false.

The general structure of Modus Tollens is:

• If \(p \rightarrow q\) is true (if \(p\) implies \(q\)),
• and \(eg q\) is true (\(q\) is false),
• then it must be the case that \(eg p\) is true (\(p\) is false).

For example, we had a statement \(\sim p \vee r\) and \(\sim r\). By using Modus Tollens, we determined that \(\sim(\sim p \vee r)\) must be true, allowing us to further simplify and analyze logical relationships.

Understanding Modus Tollens is crucial because it helps in concluding the truth of premises and rejecting false implications.

De Morgan's Law

De Morgan's Laws are essential rules used to negate expressions that contain "and" and "or" operators. They allow simplifying complex logical expressions.

These laws are particularly useful in dealing with disjunctions and conjunctions. Here's how De Morgan's Laws apply:

• The negation of a conjunction: \(\sim (p \wedge q)\) is equivalent to \(\sim p \vee \sim q\).
• The negation of a disjunction: \(\sim (p \vee q)\) is equivalent to \(\sim p \wedge \sim q\).

In our example, we used De Morgan's Law to transform \(\sim(\sim p \vee r)\) into \(p \wedge \sim r\). This transformation facilitated further deductions and conclusions in the logical argument we're solving.

De Morgan's Laws are handy tools for breaking down complex statements into simpler forms that are easier to handle in logical reasoning.

Modus Ponens

Modus Ponens is a foundational rule of inference in logic. It enables us to conclude the truth of a statement if specific conditions are met.

The pattern of Modus Ponens is as follows:

• Start with a conditional statement \(p \rightarrow q\).
• If \(p\) is proven true, then \(q\) can be concluded to be true.

In our argument, once we established that \(p\) is true from earlier deductions, we used Modus Ponens with the conditional statement \(p \rightarrow q\) to logically arrive at the solution that \(q\) must also be true.

Understanding Modus Ponens is crucial for logical reasoning as it allows one to draw conclusions from given conditions effectively.