Logic and Propositional Calculus Calculator

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Example

Created on 2024-06-20Asked by Samuel Baker (Solvelet student)

Determine the truth value of the logical statement

(p \land q) \lor (\neg p \land r)

given

p = \text{true}

q = \text{false}

, and

r = \text{true}

Solution

Given the logical statement

(p \land q) \lor (\neg p \land r)

and the truth values

p = \text{true}

q = \text{false}

, and

r = \text{true}

, we evaluate the statement step-by-step. 1. Evaluate

p \land q

p \land q = \text{true} \land \text{false} = \text{false}

2. Evaluate

\neg p

\neg p = \neg (\text{true}) = \text{false}

3. Evaluate

\neg p \land r

\neg p \land r = \text{false} \land \text{true} = \text{false}

4. Combine the results using the disjunction

\lor

(p \land q) \lor (\neg p \land r) = \text{false} \lor \text{false} = \text{false}

Therefore, the truth value of the logical statement

(p \land q) \lor (\neg p \land r)

is:

\text{false}

Solved on Solvelet with Basic AI Model

Some of the related questions asked by William Anderson on Solvelet

1. Determine the truth value of the compound statement

p \lor (q \land \neg r)

given that

p

is false,

q

is true, and

r

is false.2. Construct a truth table for the statement '

p \rightarrow (q \lor r)

' and determine its logical equivalence to '

\neg p \lor (q \land r)

DefinitionLogic and propositional calculus are branches of mathematics. & Philosophy that study formal systems of reasoning. They contain statements (propositions) that take the values either true or false and are combined with logical connectives such as AND, OR, NOT, and IMPLIES. Propositional calculus (also known as propositional logic) has expressions to denote logical statements and inference rules to deduce new raw statements. In propositional logic, the statement p∧q will imply r means “if p and q both are true then r is true.

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