Problem 33 - 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. $$p \wedge \sim p \equiv f$$

Step-by-Step Solution

Answer

The given statement \(p \wedge \sim p \equiv f\) is TRUE, as the compound statement \(p \wedge \sim p\) is False in both cases, making it equivalent to a contradiction (f).

Step 1: Analyze the given statement

We are given the statement: $$ p \wedge \sim p \equiv f $$ It is a compound statement involving conjunction (AND) and negation (NOT). In this case, we have the statement p AND the negation of p, and we want to determine if it is equivalent to a contradiction (f).

Step 2: Truth table of p AND NOT p

Let's construct a truth table for the statement \(p \wedge \sim p\): | p | NOT p | p AND NOT p | |---|-------|------------| | T | F | F | | F | T | F |

Step 3: Compare the resulting compound statement to a contradiction

We can observe that the compound statement \(p \wedge \sim p\) is False in both cases. This means that it is equivalent to a contradiction (f). Therefore, the given statement is true. To Summarize: The given statement \(p \wedge \sim p \equiv f\) is TRUE.

Key Concepts

Tautology

A tautology is a statement in propositional logic that is always true, regardless of the truth values of its individual components. Tautologies are important in logical deductions and proofs because they represent statements or propositions that are universally true.
Understanding tautologies means recognizing statements where every possible combination of true or false for the components leads to a true result. Consider the statement "The sky is blue or the sky is not blue." This example is simple but illustrates a key principle: no matter what the real state of the sky might be, it's always going to be true that the sky is either blue or not blue.

• Useful in verifying logical arguments
• Always true across any truth assignments
• Support in constructing logical proofs

Using tautologies can streamline complex arguments, providing a bedrock of certainty because these propositions don't change their truth value. Recognizing tautologies involves practice and familiarity with logical operators such as AND, OR, and NOT.

Contradiction

A contradiction in propositional logic is a statement that is always false, no matter what truth values are assigned to its individual components. Much like tautologies, contradictions are pivotal in logical arguments as they represent statements that can never be true.
A simple example of a contradiction is the statement "It is raining and it is not raining." Clearly, this scenario can never be true simultaneously—it defies reality. Thus, understanding contradictions involves identifying scenarios where logical combinations annul each other.

• Identifying contradictions helps avoid logical errors
• Always false under any circumstances
• Useful in proving the falsity of an argument or assumption by contradiction

Recognizing contradictions is crucial when analyzing arguments, as the presence of one signals that there is a logical inconsistency somewhere in the assumptions or deductions.

Truth Table

Truth tables are fundamental tools in propositional logic, used to analyze logical statements and their components systematically. They help in determining the truth value of a proposition based on all possible combinations of truth values for its components.
To create a truth table, you list all possible truth values each variable can take. For a single variable, there are two possibilities: true or false. With two variables, there are four combinations (TT, TF, FT, FF), and so on.
The columns of a truth table represent:

• Each individual variable (e.g., p, q)
• The negation of variables, if used (e.g., NOT p)
• The compound operations (e.g., p AND q, p OR q)

For example, when analyzing the statement \(p \wedge \sim p\), a truth table would show that for each possible value of p (True or False), the statement results in false. This demonstrates its status as a contradiction.
Truth tables simplify understanding logical statements by displaying all path outcomes, aiding in the identification of tautologies, contradictions, and valid arguments.