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Questions and Answers
What is the primary symbol used to represent a conjunction in logic?
What is the primary symbol used to represent a conjunction in logic?
- ∧ (correct)
- ∨
- ↔
- →
Which of the following statements correctly represents De Morgan's Law for disjunction?
Which of the following statements correctly represents De Morgan's Law for disjunction?
- ∼(p ∨ q) ∴ ∼p ∨ ∼q
- ∼(p ∧ q) ∴ ∼p ∨ ∼q
- ∼(p ∧ q) ∴ ∼p ∧ ∼q
- ∼(p ∨ q) ∴ ∼p ∧ ∼q (correct)
If the premises are p → q and extasciitilde q, what conclusion can be drawn using Modus Tollens?
If the premises are p → q and extasciitilde q, what conclusion can be drawn using Modus Tollens?
- q
- p
- extasciitilde p (correct)
- None of the above
What is the conclusion of the law of disjunctive inference given the premises p ∨ q and extasciitilde p?
What is the conclusion of the law of disjunctive inference given the premises p ∨ q and extasciitilde p?
How is a biconditional statement expressed in symbolic form?
How is a biconditional statement expressed in symbolic form?
Which truth table condition for p and q results in p ∧ q being true?
Which truth table condition for p and q results in p ∧ q being true?
In the context of conditionals, which of the following statements is logically equivalent to the contrapositive of p → q?
In the context of conditionals, which of the following statements is logically equivalent to the contrapositive of p → q?
What is a tautology in logical terms?
What is a tautology in logical terms?
Which of the following statements accurately describes the relationship between a conditional statement and its contrapositive?
Which of the following statements accurately describes the relationship between a conditional statement and its contrapositive?
Under which condition is a biconditional statement p ↔ q true?
Under which condition is a biconditional statement p ↔ q true?
What is the conclusion derived from the premises p → q and p using the Law of Modus Ponens?
What is the conclusion derived from the premises p → q and p using the Law of Modus Ponens?
Identify the correct conclusion drawn from the following premises using the Law of Simplification: p ∧ q.
Identify the correct conclusion drawn from the following premises using the Law of Simplification: p ∧ q.
According to De Morgan's Law, what is the conclusion for the premise ¬(p ∧ q)?
According to De Morgan's Law, what is the conclusion for the premise ¬(p ∧ q)?
What conclusion can be drawn from the premises p ∨ q and ¬q using the Law of Disjunctive Inference?
What conclusion can be drawn from the premises p ∨ q and ¬q using the Law of Disjunctive Inference?
Which of the following is true regarding a tautology in logic?
Which of the following is true regarding a tautology in logic?
Which statement accurately portrays the process of logical equivalence?
Which statement accurately portrays the process of logical equivalence?
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Study Notes
Logic Exam Topics
- Mathematical Sentences: Defined expressions that can be true or false.
- Nonmathematical Sentences: Statements that do not fit traditional mathematical criteria.
- Open Sentences: Sentences containing variables, whose truth value is not constant.
- Closed Sentences: Sentences without variables, always true or false.
Negation
- Symbol: ~
- Represents the opposite truth value of a given statement.
Conjunctions (And)
- Symbol: ∧
- Evaluates to true only when both statements are true:
- Truth Table:
- T ∧ T = T
- T ∧ F = F
- F ∧ T = F
- F ∧ F = F
- Truth Table:
Disjunctions (Or)
- Symbol: ∨
- Evaluates to true when at least one statement is true:
- Truth Table:
- T ∨ T = T
- T ∨ F = T
- F ∨ T = T
- F ∨ F = F
- Truth Table:
Conditionals (If..., then...)
- Symbol: →
- False only when the first statement is true and the second is false:
- Truth Table:
- T → T = T
- T → F = F
- F → T = T
- F → F = T
- Truth Table:
Biconditionals (if and only if)
- Symbol: ↔
- True when both statements are either true or false:
- Truth Table:
- T ↔ T = T
- T ↔ F = F
- F ↔ T = F
- F ↔ F = T
- Truth Table:
Tautology
- Defined as a compound statement that is always true, regardless of individual truth values.
Logically Equivalent Statements
- Two statements are logically equivalent if they have identical truth values in all scenarios.
De Morgan's Law
- For conjunctions: ~(p ∧ q) = ~p ∨ ~q
- For disjunctions: ~(p ∨ q) = ~p ∧ ~q
Conditionals and Related Terms
- Conditional: p → q
- Converse: q → p
- Inverse: ~p → ~q
- Contrapositive: ~q → ~p
- Conditionals and contrapositives are logically equivalent.
Law of Contrapositives
- Based on a conditional p → q, states that if q is false, then p must also be false: ~q → ~p
Law of Modus Ponens (Law of Detachment)
- If p → q is true and p is true, then q must be true.
Law of Modus Tollens
- If p → q is true and q is false (~q), then p must also be false (~p).
Law of Disjunctive Inference
- If p ∨ q is true and one of the statements is false (~p or ~q), then the other must be true.
Law of Conjunction
- If both p and q are true, then p ∧ q holds true.
Law of Simplification
- If p ∧ q is true, either p or q can be regarded as true individually.
Law of Disjunctive Addition
- If p is true, then it can be inferred that p ∨ q is also true.
Chain Rule (Law of Syllogism)
- Connects two conditionals: if p → q and q → r are both true, then p → r is also true.
Logic Exam Topics
- Mathematical Sentences: Defined expressions that can be true or false.
- Nonmathematical Sentences: Statements that do not fit traditional mathematical criteria.
- Open Sentences: Sentences containing variables, whose truth value is not constant.
- Closed Sentences: Sentences without variables, always true or false.
Negation
- Symbol: ~
- Represents the opposite truth value of a given statement.
Conjunctions (And)
- Symbol: ∧
- Evaluates to true only when both statements are true:
- Truth Table:
- T ∧ T = T
- T ∧ F = F
- F ∧ T = F
- F ∧ F = F
- Truth Table:
Disjunctions (Or)
- Symbol: ∨
- Evaluates to true when at least one statement is true:
- Truth Table:
- T ∨ T = T
- T ∨ F = T
- F ∨ T = T
- F ∨ F = F
- Truth Table:
Conditionals (If..., then...)
- Symbol: →
- False only when the first statement is true and the second is false:
- Truth Table:
- T → T = T
- T → F = F
- F → T = T
- F → F = T
- Truth Table:
Biconditionals (if and only if)
- Symbol: ↔
- True when both statements are either true or false:
- Truth Table:
- T ↔ T = T
- T ↔ F = F
- F ↔ T = F
- F ↔ F = T
- Truth Table:
Tautology
- Defined as a compound statement that is always true, regardless of individual truth values.
Logically Equivalent Statements
- Two statements are logically equivalent if they have identical truth values in all scenarios.
De Morgan's Law
- For conjunctions: ~(p ∧ q) = ~p ∨ ~q
- For disjunctions: ~(p ∨ q) = ~p ∧ ~q
Conditionals and Related Terms
- Conditional: p → q
- Converse: q → p
- Inverse: ~p → ~q
- Contrapositive: ~q → ~p
- Conditionals and contrapositives are logically equivalent.
Law of Contrapositives
- Based on a conditional p → q, states that if q is false, then p must also be false: ~q → ~p
Law of Modus Ponens (Law of Detachment)
- If p → q is true and p is true, then q must be true.
Law of Modus Tollens
- If p → q is true and q is false (~q), then p must also be false (~p).
Law of Disjunctive Inference
- If p ∨ q is true and one of the statements is false (~p or ~q), then the other must be true.
Law of Conjunction
- If both p and q are true, then p ∧ q holds true.
Law of Simplification
- If p ∧ q is true, either p or q can be regarded as true individually.
Law of Disjunctive Addition
- If p is true, then it can be inferred that p ∨ q is also true.
Chain Rule (Law of Syllogism)
- Connects two conditionals: if p → q and q → r are both true, then p → r is also true.
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