Question

Show that the implication "if \(\mathrm{a}\), then \(\mathrm{b}^{\prime \prime}\) and the disjunction "(not a) or \(\mathrm{b}^{\prime \prime}\) are logically equivalent.

Short answer

By constructing truth tables for the implication "if a, then b''" and the disjunction "(not a) or b''" and evaluating their truth values for all possible combinations of a and b'', we find that their truth tables are identical in all cases. Therefore, the implication "if a, then b''" and the disjunction "(not a) or b''" are logically equivalent.

Step-by-step solution

Construct Truth Tables

We will start by constructing the truth table for the implication "if a, then b''" and for the disjunction "(not a) or b''". We need to consider all possible truth value combinations for a and b''. Here are the possible combinations: 1. a: true, b'': true 2. a: true, b'': false 3. a: false, b'': true 4. a: false, b'': false

Evaluate Implication and Disjunction

Now, we'll evaluate the implication "if a, then b''" and disjunction "(not a) or b''" given the truth values for each combination. 1. Implication (if a, then b''): true -> true = true; Disjunction (not a or b''): (not true) or true = true 2. Implication (if a, then b''): true -> false = false; Disjunction (not a or b''): (not true) or false = false 3. Implication (if a, then b''): false -> true = true; Disjunction (not a or b''): (not false) or true = true 4. Implication (if a, then b''): false -> false = true; Disjunction (not a or b''): (not false) or false = true

Compare Truth Tables

Comparing the truth tables for the implication "if a, then b''" and the disjunction "(not a) or b''", we can find the following: 1. Implication: true; Disjunction: true 2. Implication: false; Disjunction: false 3. Implication: true; Disjunction: true 4. Implication: true; Disjunction: true As we can see, the truth tables for both expressions are identical in all cases. Thus, we can conclude that the implication "if a, then b''" and the disjunction "(not a) or b''" are logically equivalent.

Key concepts

Truth Tables

Truth tables are essential tools in logic and computer science, used to determine the truth value of complex logical expressions based on the truth values of their individual components. They list all possible combinations of truth values for given logical variables and show how these combine to form the overall truth value of an expression.

For example, consider the logical statement 'if a, then b'. To examine this using a truth table, we assign truth values (True or False) to 'a' and 'b'. Each possibility is considered: when both a and b are True, when a is True and b is False, when a is False and b is True, and when both are False. The truth value of the entire statement for each scenario is listed, helping us grasp how changes in variables affect the result.

Implication

In logic, an implication, often expressed as 'if a, then b' or written as 'a implies b', is a type of logical operation. If the first part (the antecedent 'a') is True, then the second part (the consequent 'b') must also be true for the statement to hold. However, if 'a' is False, the implication remains True irrespective of the truth value of 'b'.

Truth Values of Implication

Creating a truth table for the implication helps us see that if 'a' is True and 'b' is True, the implication is True. But if 'a' is True and 'b' is False, the implication is False. With 'a' False (whether 'b' is True or False), the implication is True. This may initially seem counterintuitive to some but is a fundamental aspect of formal logic.

Disjunction

A disjunction is another fundamental logical operator, commonly known as 'or'. It is denoted as 'a or b' and is True if at least one of 'a' or 'b' is True. If both are False, then the disjunction is False. In the context of logical equivalence, we often deal with an inclusive disjunction, which means that 'a or b' is True even if both 'a' and 'b' are True.

Negation in Disjunction

The exercise involves a special form of disjunction that includes negation: '(not a) or b'. Here, the disjunction is True if 'a' is False, or 'b' is True, or both conditions are met. The negation flips the truth value of 'a', changing the dynamic of the disjunction.

Logical Operators

Logical operators are symbols or words used to connect propositions or statements in a way that reflects their logical relationship. These include 'and' (conjunction), 'or' (disjunction), 'not' (negation), 'if...then...' (implication), and 'if and only if' (biconditional). Each operator has rules that dictate the truth values of statements based on the truth values of the components involved.

Understanding these operators and their interactions is crucial in evaluating arguments, programming, and various fields of mathematics. When studying logical equivalence, you often compare the truth tables created by different combinations of logical operators to determine if they yield the same result in all possible scenarios.