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Questions and Answers
What is the primary task in HW3 as described in the content?
What is the primary task in HW3 as described in the content?
- Show that several logical expressions are equivalent. (correct)
- Write new logical expressions from given axioms.
- Prove the validity of a single logical statement.
- Identify the components of logical expressions.
Which logical operation is likely involved in showing equivalence of expressions as stated?
Which logical operation is likely involved in showing equivalence of expressions as stated?
- Implication (IF...THEN)
- Conjunction (AND) (correct)
- Negation (NOT)
- Disjunction (OR)
What format do logical expressions generally adhere to in such problems?
What format do logical expressions generally adhere to in such problems?
- List of conditions to fulfill.
- A sentence form expressing a proposition.
- Numerical equality like $x = 5$.
- Symbolic representation using logical operators. (correct)
When proving that logical expressions are equivalent, which concept is key?
When proving that logical expressions are equivalent, which concept is key?
Why might equivalence in logical expressions be significant in logic?
Why might equivalence in logical expressions be significant in logic?
Flashcards
Logical Equivalence
Logical Equivalence
Two logical expressions are equivalent if they produce the same truth value for all possible combinations of truth values of their variables.
Truth Values
Truth Values
The possible values (true or false) a logical expression can take.
Logical Expressions
Logical Expressions
Statements that can be evaluated as true or false.
HW3 Task
HW3 Task
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Show equivalence
Show equivalence
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Study Notes
Important Logical Equivalences
- Double Negation: ¬¬p <=> p
- Commutative: p∨q <=> q∨p AND p∧q <=> q∧p
- Associative: (p∨q)∨r <=> p∨(q∨r) AND (p∧q)∧r <=> p∧(q∧r)
- Identity: p∧T <=> p AND p∨F <=> p
- Domination: p∨T <=> T AND p∧F <=> F
- Idempotent: p∨p <=> p AND p∧p <=> p
- Distributive: p∨(q∧r) <=> (p∨q)∧(p∨r) AND p∧(q∨r) <=> (p∧q)∨(p∧r)
- De Morgan's: ¬(p∨q) <=> ¬p∧¬q AND ¬(p∧q) <=> ¬p∨¬q
- Other Useful Equivalences: p∨¬p <=> T AND p∧¬p <=> F AND p↔q <=> (¬p∨q)∧(p∨¬q)
Using Logical Equivalences in Proofs
- Equivalences can transform propositions or parts of propositions to reach a conclusion.
- Example: ((p∧q)→p) is a tautology.
- Proof: ((p∧q)→p) <=> ¬(p∧q)∨p <=> (¬p∨¬q)∨p <=> ¬q∨(¬p∨p) <=> ¬q∨T <=> T
Truth Table Approach vs. Logical Equivalence
- Truth tables are exhaustive but become impractical for many propositions.
- Logical equivalences provide an alternative, often faster approach.
Predicate Logic
- Constants: represent specific objects (e.g., John, 7)
- Variables: represent objects of a specific type (defined by the universe of discourse).
- Predicates: represent relations or properties among objects (e.g., greater than, student, married).
- Predicates are not propositions until variables are substituted with specific values, creating propositions (e.g., John > Peter is a proposition).
- Predicate logic remedies propositional logic's limitations by explicitly modeling objects and properties, allowing statements with variables and quantification.
Quantified Statements
- Universal Quantification (∀): ∀x P(x) means P(x) is true for every x in the universe of discourse.
- Existential Quantification (∃): ∃x P(x) means P(x) is true for at least one x in the universe of discourse.
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Description
This quiz explores important logical equivalences fundamental to propositional logic. It covers concepts like double negation, commutative properties, and De Morgan's laws along with their applications in proofs and truth tables. Test your understanding of these principles and their usefulness in logical reasoning.
