Problem 29 - Discrete Mathematics with Applications

Question

Mark each sentence as true or false, where \(p, q,\) and \(r\) are arbitrary statements, \(t\) a tautology, and \(f\) a contradiction. $$p \vee q \equiv q \vee p$$

Step-by-Step Solution

Answer

The given expression \(p \vee q = q \vee p\) is a tautology, as it holds true for all possible combinations of truth values for \(p\) and \(q\). Therefore, the answer is true.

Step 1: Create a truth table

First, create a truth table for the given expression. We will list all possible combinations of truth values for \(p\) and \(q\), and then evaluate the result of the expression for each combination.

Step 2: Evaluate the expression for each combination

Now, apply the OR operation on \(p\) and \(q\) for each combination and check if the result is equal on both sides of the equation. \(p \vee q\) is true if at least one of the propositions \(p\) or \(q\) is true. Otherwise, it is false. Here is the truth table showing the values of \(p \vee q\) and \(q \vee p\) for each combination: | \(p\) | \(q\) | \(p \vee q\) | \(q \vee p\) | |------|------|------------|------------| | T | T | T | T | | T | F | T | T | | F | T | T | T | | F | F | F | F |

Step 3: Verify if \(p \vee q = q \vee p\) for all the combinations

Observe that in the truth table, the values of \(p \vee q\) and \(q \vee p\) are equal for each possible combination of truth values for \(p\) and \(q\).

Step 4: Conclusion

Since \(p \vee q = q \vee p\) for all the combinations, the given expression is true (a tautology). Therefore, we can mark the sentence as true.

Key Concepts

Truth Tables

Truth tables are essential tools in discrete mathematics for understanding logical expressions. They allow us to systematically explore all possible truth values of logical variables. For any combination of these variable truth values, we can determine the truth value of a more complex expression.

Here's how it works:

• Identify all the variables in the expression. For example, in the expression \( p \vee q \), the variables are \( p \) and \( q \).
• List all possible combinations of truth values for these variables. If you have two variables, like \( p \) and \( q \), there will be four combinations: True-True, True-False, False-True, and False-False.
• Evaluate the expression using each combination. Place the results in the corresponding row to check consistency across different expressions or sides of an equation.

By examining these results, we can make conclusions about logical equivalences or other properties, making truth tables a powerful method for logical analysis.

Logical Expressions

Logical expressions are statements that use logical operators to connect smaller propositions. These operators, such as AND (\(\wedge\)), OR (\(\vee\)), and NOT (\(eg\)), help us form complex reasoning structures from simpler components.

• OR (\(\vee\)): The expression \( p \vee q \) is true if either \( p \) is true, \( q \) is true, or both are true. It only becomes false when both propositions are false.
• AND (\(\wedge\)): The expression \( p \wedge q \) is true only when both \( p \) and \( q \) are true. If either is false, then the entire expression is false.
• NOT (\(eg\)): The expression \( eg p \) negates the truth value of \( p \). So if \( p \) is true, \( eg p \) is false, and vice versa.

By understanding and mastering these operators, we can construct, evaluate, and simplify logical statements across various scenarios in mathematics and computer science.

Tautology

A tautology is a logical expression that is true in every possible scenario. No matter what the values of the individual propositions are, the expression will always evaluate to true.

This concept is crucial because it helps us identify expressions that are universally valid, akin to laws in logic. For instance, one simple tautology is \( p \vee eg p \), which occurs because one of these will always hold true. Thus, in any situation, such an expression remains true, regardless of the truth values of its constituent variables.

Identifying a tautology, like in our exercise with \( p \vee q = q \vee p \), reinforces the commutative property of the OR operator, demonstrating inherent symmetry and predictability.

Commutative Property

The commutative property is a fundamental principle in logic, mathematics, and algebra. It states that the order of applying a certain operator does not affect the outcome. For logical operators, this property often applies to AND and OR.

For example, in logical expressions:

• \( p \vee q = q \vee p \): The OR operator is commutative. It means changing the order does not change the outcome.
• \( p \wedge q = q \wedge p \): Similarly, the AND operator also upholds the commutative property.

Understanding this property allows us to simplify and manipulate expressions without altering their truth values. It proves helpful in various aspects of problem-solving, ensuring reliability in logical equations and equivalence transformations.