Question

Construct a truth table for \([(\mathrm{a} \vee \mathrm{b}) \wedge \sim \mathrm{b}] \rightarrow \mathrm{a}\).

Short answer

The short answer for the truth table of the given logical expression [(a ∨ b) ∧ ¬b] → a is: | a | b | ¬b | (a ∨ b) | [(a ∨ b) ∧ ¬b] | [(a ∨ b) ∧ ¬b] → a | |---|---|----|--------|--------------|---------------------| | F | F | T | F | F | T | | F | T | F | T | F | T | | T | F | T | T | F | T | | T | T | F | T | F | T |

Step-by-step solution

List all possible truth values for a and b

There are 2 variables in the expression: a and b. Each variable can have 2 possible truth values: True (T) or False (F). Therefore, we have 4 combinations of truth values for a and b.

Compute ¬b for all possible values of b

We need to find the truth values for ¬b, given the truth values of b. This will be used later in the expression. For each row in the truth table, apply the NOT operation to the value of b: ¬F = T ¬T = F

Compute (a ∨ b) for all possible truth value combinations of a and b

Now, we apply the OR operation to the truth value combinations of a and b. F ∨ F = F F ∨ T = T T ∨ F = T T ∨ T = T

Compute [(a ∨ b) ∧ ¬b] for all possible combinations of (a ∨ b) and ¬b

Now for each corresponding truth value output for (a ∨ b) from Step 3 and ¬b from Step 2, we apply the AND operation as follows: (F ∧ T) = F (T ∧ F) = F (T ∧ F) = F (T ∧ F) = F

Compute [(a ∨ b) ∧ ¬b] → a for all possible combinations of [(a ∨ b) ∧ ¬b] and a

Now, we apply the implication (→) operation to our computed truth values of [(a ∨ b) ∧ ¬b] from Step 4 and the original values of a from Step 1: F → F = T F → F = T F → T = T F → T = T

Assemble the truth table for [(a ∨ b) ∧ ¬b] → a

Now we can put all the results from Steps 1-5 to construct the truth table for the given expression as follows: | a | b | ¬b | (a ∨ b) | [(a ∨ b) ∧ ¬b] | [(a ∨ b) ∧ ¬b] → a | |---|---|----|--------|--------------|---------------------| | F | F | T | F | F | T | | F | T | F | T | F | T | | T | F | T | T | F | T | | T | T | F | T | F | T |

Key concepts

Logical Connectives

Logical connectives are the backbone of propositional logic. They are symbols or words used to connect two or more statements, yielding a single logical expression with a truth value that depends on the truth values of its components. Common logical connectives include AND (\(\bigwedge\)), OR (\(\vee\)), NOT (\(\sim\)), IMPLICATION (\(\rightarrow\)), and EQUIVALENCE (\(\leftrightarrow\)).

For instance, the AND connective (\(\wedge\)) requires both propositions it connects to be true for the whole expression to be true, while the OR connective (\(\vee\)) requires only one to be true. The NOT connective (\(\sim\)) flips the truth value of a single proposition, turning true into false, and vice versa. IMPLICATION (\(\rightarrow\)) can be trickier: a proposition \(p \rightarrow q\) is only false if \(p\) is true and \(q\) is false; otherwise, it is true. Understanding how these work is crucial for constructing truth tables and analyzing logical statements.

Propositional Logic

Propositional logic, also known as sentential logic, is a branch of logic that studies ways of joining and/or modifying entire propositions, statements or sentences to form more complicated propositions, statements or sentences. A proposition in this context is a statement that is either true or false but not both—a declarative statement with a truth value.

In propositional logic, simple statements can be represented by symbols (commonly lowercase letters such as \(a\), \(b\), and so on) and combined using logical connectives to form more complex expressions. The beauty of this system lies in its ability to break down complex arguments into simpler components that can be analyzed for validity. The construction of a truth table is a practical application of propositional logic that displays all the possible truth values of a logical expression.

Truth Values

Truth values are the foundation of logic, representing the state of propositions as either 'true' or 'false'. In the realm of propositional logic, we evaluate the truth or falsity of each component of a logical expression to determine the expression’s overall truth value.

When constructing truth tables, each column corresponds to a component of the logical expression, and rows represent possible combinations of truth values for these components. By systematically working through these combinations and applying the relevant logical connectives, we can ascertain the truth value of complex logical expressions in every possible scenario. This exercise in unambiguous assignment of truth values is fundamental in analyzing logical arguments, debugging digital logic circuits, and programming conditional statements in computer science.