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Chapter 3: Problem 23

Construct a truth table for the given statement. \((\sim p \leftrightarrow q) \rightarrow(\sim p \rightarrow q)\)

Short Answer

Expert verified
The truth table for the given statement is as follows: \n\np = T, q = T -> \( (\sim p \leftrightarrow q) \rightarrow (\sim p \rightarrow q)\) = T\n\np = T, q = F -> \( (\sim p \leftrightarrow q) \rightarrow (\sim p \rightarrow q)\) = F\n\np = F, q = T -> \( (\sim p \leftrightarrow q) \rightarrow (\sim p \rightarrow q)\) = T\n\np = F, q = F -> \( (\sim p \leftrightarrow q) \rightarrow (\sim p \rightarrow q)\) = T

Step by step solution

01

Understand the Logical Operators

The logical statement contains these operators: \n\n1. Negation (\sim) - When the proposition is 'true', the negation is 'false'. When the proposition is 'false', the negation is 'true'. \n\n2. Bi-conditional (\leftrightarrow) - The statement is 'true' if both propositions have the same truth value, and 'false' otherwise. \n\n3. Conditional (\rightarrow) - If the first proposition (also known as antecedent) is 'true' and the second proposition (consequence) is 'false', the whole statement is 'false'. In all other cases, the statement is 'true'.
02

List all Possible Combinations of Propositions

The logical statement deals with two propositions 'p' and 'q'. As both can be either true or false, four combinations are possible: \n\n1. p = true, q = true\n2. p = true, q = false\n3. p = false, q = true\n4. p = false, q = false
03

Evaluate the Innermost Statements

Find the truth values of the innermost statements first. Here, consider \(\sim p\) (negation of p) and \( \sim p \leftrightarrow q\) (negation of p bi-conditional q). Calculate these based on the truth values of p and q from Step 2.
04

Evaluate the Entire Statement

Now, using results of Step 3, calculate the truth values of the entire statement \( (\sim p \leftrightarrow q) \rightarrow (\sim p \rightarrow q)\). Apply the rule of Conditional operator.

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