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Questions and Answers
Which term refers to a proposition that follows directly from a proved theorem?
Which term refers to a proposition that follows directly from a proved theorem?
- Conjecture
- Lemma
- Fallacy
- Corollary (correct)
What is required for a correct deductive proof?
What is required for a correct deductive proof?
- It does not require any conditions to be true.
- If the conditions are true, then the conclusion must be true. (correct)
- It must consist of more than three statements.
- All conditions must be false for the conclusion to be false.
Which of the following describes a conjecture?
Which of the following describes a conjecture?
- A statement whose truth value is unknown until proven. (correct)
- A simple theorem used as a step in proving another theorem.
- A case of incorrect reasoning.
- A statement that has been proven true.
What is indicated by the statement 'A is a subset of B'?
What is indicated by the statement 'A is a subset of B'?
What does set equality mean?
What does set equality mean?
What type of quantifier indicates that a statement is true for at least one value?
What type of quantifier indicates that a statement is true for at least one value?
Which of the following is an example of a contradiction?
Which of the following is an example of a contradiction?
Which law states that combining a variable with itself results in the variable itself?
Which law states that combining a variable with itself results in the variable itself?
What do De Morgan's Laws relate to?
What do De Morgan's Laws relate to?
Which statement illustrates the Implication Law?
Which statement illustrates the Implication Law?
What type of proposition is neither a tautology nor a contradiction?
What type of proposition is neither a tautology nor a contradiction?
Which of the following is NOT an Identity Law?
Which of the following is NOT an Identity Law?
In logical equivalence, if two propositions are considered equal, what is true about their truth values?
In logical equivalence, if two propositions are considered equal, what is true about their truth values?
What does the Absorption Law state regarding logical operations?
What does the Absorption Law state regarding logical operations?
Which of the following statements best describes a tautology?
Which of the following statements best describes a tautology?
Flashcards
Existential Quantifier
Existential Quantifier
A quantifier that means 'at least one' or 'some'.
Universal Quantifier
Universal Quantifier
A quantifier that means 'all' or 'every'.
Tautology
Tautology
A statement always true, regardless of input.
Contradiction
Contradiction
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Contingency
Contingency
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Logical Equivalence
Logical Equivalence
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Identity Law
Identity Law
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Commutative Law
Commutative Law
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De Morgan's Law
De Morgan's Law
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Implication Law
Implication Law
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Proof
Proof
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Theorem
Theorem
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Set
Set
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Subset
Subset
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Set Equality
Set Equality
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Study Notes
Quantifiers
- Quantifiers are words that indicate how many examples exist.
- Existential quantifiers indicate at least one example. Examples include "some," "sometimes," "there is," and "there exists."
- Universal quantifiers indicate all examples. Examples include "all," "always," "none," and "never."
Propositional Equivalences
- Tautology is a statement that is always true.
- Contradiction is a statement that is always false.
- Contingency is a compound proposition that is neither a tautology nor a contradiction.
Logical Equivalence
- Logically equivalent propositions have the same truth values in all cases.
Equivalence Laws
- Identity Laws state that, for any proposition p, p ∧ T = p and p ∨ F = p.
- Domination Laws state that, for any proposition p, p ∨ T = T and p ∧ F = F.
- Double Negation Law states that for any proposition p, ~ (~p) = p.
- Idempotent Laws state that p ∧ p = p and p ∨ p = p.
- Commutative Laws state that p ∨ q = q ∨ p and p ∧ q = q ∧ p.
Associative Laws
- (p ∨ q) ∨ r = p ∨ (q ∨ r)
- (p ∧ q) ∧ r = p ∧ (q ∧ r)
Distributive Laws
- p ∨ (q ∧ r) = (p ∨ q) ∧ (p ∨ r)
- p ∧ (q ∨ r) = (p ∧ q) ∨ (p ∧ r)
De Morgan's Laws
- ~(p ∧ q) = ~p ∨ ~q
- ~(p ∨ q) = ~p ∧ ~q
Absorption Laws
- p ∨ (p ∧ q) = p
- p ∧ (p ∨ q) = p
Negation Laws
- p ∨ ~p = T
- p ∧ ~p = F
Implication Law
- p → q = ~p ∨ q
Biconditional Property
- p ↔ q = (p ∧ q) ∨ (~p ∧ ~q)
Contrapositive of Conditional Statement
- p → q = ~q → ~p
Axioms
- Fundamentals truths or proven statements.
Proof
- Demonstrates the truth of a statement using a sequence ofstatements based on axioms or previously proven facts.
Set Theory
- Sets are collections of objects called elements.
- Order of elements in a set does not matter.
- Repetition of elements is not counted.
Set Equality
- Sets A and B are equal if and only if they contain exactly the same elements.
Subset
- A is a subset of B (denoted A ⊆ B) if every element of A is also an element of B.
Rules of Inference
- These are steps that connect statements.
Fallacies
- Incorrect reasoning.
Theorem
- A provable fact.
Lemma
- A small theorem used to prove a larger theorem.
Corollary
- A statement that follows directly from a theorem.
Conjecture
- A statement whose truth is unproven.
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