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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?
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?
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?
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?
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How is a biconditional statement expressed in symbolic form?
How is a biconditional statement expressed in symbolic form?
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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?
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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?
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What is a tautology in logical terms?
What is a tautology in logical terms?
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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?
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Under which condition is a biconditional statement p ↔ q true?
Under which condition is a biconditional statement p ↔ q true?
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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?
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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.
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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)?
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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?
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Which of the following is true regarding a tautology in logic?
Which of the following is true regarding a tautology in logic?
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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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Description
Prepare for your Geometry exam with this quiz focusing on Unit 1: Logic. You'll explore mathematical and nonmathematical sentences, negation, conjunctions, and truth tables. Test your understanding of the foundational concepts that are crucial for further studies in geometry.
