Question

Construct truth tables for the following statements. (1) \((\mathrm{a} \rightarrow \mathrm{b}) \wedge(\sim \mathrm{b} \rightarrow \sim \mathrm{a})\) (2) \(\sim(\mathrm{a} \vee \mathrm{b}) \vee \sim(\mathrm{a} \wedge \mathrm{b})\)

Short answer

The truth tables for the given statements are as follows: (1) \((a \rightarrow b) \wedge (\sim b \rightarrow \sim a)\): | a | b | a→b | ¬b | ¬a | ¬b→¬a | (a→b)∧(¬b→¬a) | |:-:|---|-----|----|----|-------|----------------| | T | T | T | F | F | T | T | | T | F | F | T | F | F | F | | F | T | T | F | T | T | T | | F | F | T | T | T | T | T | (2) \(\sim(a \vee b) \vee \sim (a \wedge b)\): | a | b | a⋁b | a⋀b | ¬(a⋁b) | ¬(a⋀b) | ¬(a⋁b)⋁¬(a⋀b) | |:-:|---|-----|-----|-------|-------|---------------| | T | T | T | T | F | F | F | | T | F | T | F | F | T | T | | F | T | T | F | F | T | T | | F | F | F | F | T | T | T |

Step-by-step solution

Identify the logical operations and variables

For both statements, we have the variables 'a' and 'b'. Two logical operations are present in each statement: (1) $(a \rightarrow b) \wedge (\sim b \rightarrow \sim a)\( involves "implies" (\)\rightarrow\() and "and" (\)\wedge$). (2) \(\sim(a \vee b) \vee \sim (a \wedge b)\) involves "or" (\(\vee\)) and "and" (\(\wedge\)).

Construct the truth tables for the individual components

We need to build the truth tables considering all possible combinations of 'a' and 'b', which are True (T) and False (F). - For implication \((a \rightarrow b)\): - 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. For other operations, we can follow the same process.

Construct the truth table for the complex statement (1)

Using the truth table for individual components, we can create the truth table for the complex statement (1) \((a \rightarrow b) \wedge (\sim b \rightarrow \sim a)\): - Calculate the values of \((a \rightarrow b)\) and \((\sim b \rightarrow \sim a)\) for each combination of 'a' and 'b'. - Compute the result of \((a \rightarrow b) \wedge (\sim b \rightarrow \sim a)\) for these combinations using the "and" (\(\wedge\)) operation.

Construct the truth table for the complex statement (2)

Similarly, we can construct the truth table for the complex statement (2) \(\sim (a \vee b) \vee \sim (a \wedge b)\): - Calculate the values of \((a \vee b)\) and \((a \wedge b)\) for each combination of 'a' and 'b'. - Apply the negation (\(\sim\)) to these values. - Compute the result of \(\sim (a \vee b) \vee \sim (a \wedge b)\) for these combinations using the "or" (\(\vee\)) operation.

Key concepts

Logical Operations

Logical operations are the foundation of how we perform reasoning in various fields, including mathematics, computer science, and philosophy. They are essentially the rules that govern the manipulation of truth values in logical statements. By understanding these operations, we can evaluate the truth or falsehood of complex expressions based on the truth values of their components.

There are several basic logical operations:

• Negation ( ), which reverses the truth value of a proposition.
• Conjunction ( ), meaning 'and', where the result is true only if both propositions are true.
• Disjunction ( ), meaning 'or', where the result is true if at least one of the propositions is true.
• Implication ( ), often read as 'if...then', where the result is false only if the first proposition is true and the second is false.

Understanding how to use these operations to build and deconstruct logical statements is crucial in mastering the art of logical reasoning.

Implication in Logic

Implication in logic is a fundamental concept that is often symbolized as ( ). It connects two statements in a way that establishes a conditional relationship between them. The implication states that if the first statement (antecedent) is true, then the second statement (consequent) must also be true for the entire expression to hold true. If the antecedent is false, the implication is always true, regardless of the truth value of the consequent.

This might seem counterintuitive, but it is an accepted convention in classical logic. By stating an implication, we are not making a claim about a causal relationship, but rather defining a promise that is only broken if the first statement is true and the second statement is false. As a result, implications are critical in constructing arguments and proving theorems, where logical consistency is paramount.

Truth Table Construction

Constructing truth tables is an essential technique for analyzing and understanding the behavior of logical statements. It involves listing all possible combinations of truth values for the involved variables and then determining the truth value of the statement for each combination. A truth table provides a clear and structured way to visualize how the logical operations within a statement interact, and it is particularly useful for complex expressions involving multiple operations.

To construct a truth table:

• Start by writing down all the possible truth value combinations for the variables, typically denoted as T (True) and F (False).
• Next, calculate the truth values of the simpler components of the logical expression.
• Finally, work through the expression step by step according to the standard rules of logical operations to determine the truth value of the entire statement for each combination.

By mastering truth table construction, students enhance their ability to dissect and comprehend logical statements fully.

Logical Conjunction and Disjunction

Logical conjunction and disjunction are pivotal operations that dictate the interaction of truth values in compound propositions. Conjunction, represented by ( ), is the 'and' operation. It requires both propositions to be true for the resulting statement to be true. If either of the propositions is false, the conjunction is false.

On the other hand, disjunction, represented by ( ), is the 'or' operation, which is inclusive in nature. This means that the disjunction is true if at least one of the propositions is true. It is only false when both propositions are false. These operations help in building more complex logical expressions and are instrumental in the formulation of hypotheses, designing algorithms, and shaping logical arguments within disparate fields of study.