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Questions and Answers
What is the converse of the implication 'If two sides of a triangle are congruent then its two angles are congruent'?
What is the converse of the implication 'If two sides of a triangle are congruent then its two angles are congruent'?
If two angles of a triangle are congruent then its two sides are congruent.
What is the inverse of the implication 'If two sides of a triangle are congruent then its two angles are congruent'?
What is the inverse of the implication 'If two sides of a triangle are congruent then its two angles are congruent'?
If two sides of a triangle are not congruent then its two angles are not congruent.
What is the contrapositive of the implication 'If two sides of a triangle are congruent then its two angles are congruent'?
What is the contrapositive of the implication 'If two sides of a triangle are congruent then its two angles are congruent'?
If two angles of a triangle are not congruent then its two sides are not congruent.
What is the negation of the implication 'If two sides of a triangle are congruent then its two angles are congruent'?
What is the negation of the implication 'If two sides of a triangle are congruent then its two angles are congruent'?
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What is the existential quantifier symbol?
What is the existential quantifier symbol?
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What is the universal quantifier symbol?
What is the universal quantifier symbol?
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If a statement quantified by a universal quantifier V is true, then at least one object in the collection does not satisfy the condition.
If a statement quantified by a universal quantifier V is true, then at least one object in the collection does not satisfy the condition.
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A statement quantified by an existential quantifier is false if no object in the collection satisfies the condition.
A statement quantified by an existential quantifier is false if no object in the collection satisfies the condition.
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What is the result of the identity law for conjunction?
What is the result of the identity law for conjunction?
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What does the complement law state for a statement p?
What does the complement law state for a statement p?
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Study Notes
Logical Implications and Their Forms
- Converse of implication: If q, then p (¬q → ¬p).
- Inverse of implication: If not p, then not q (¬p → ¬q).
- Contrapositive of implication: If not q, then not p (¬q → ¬p).
Quantifiers and Quantified Statements
- Existential quantifier: Indicates "there exists", denoted by ∃.
- Universal quantifier: Indicates "for all", denoted by ∀.
- Example statement with existential quantifier: "There exists an even prime number in the set of natural numbers" relates to the set of natural numbers with the condition of being an even prime.
- Example statement with universal quantifier: "All natural numbers are positive" applies a condition to all objects in a collection.
- A statement quantified by ∀ is true if all members satisfy the condition; false if at least one does not.
- A statement quantified by ∃ is true if at least one member satisfies the condition; false if none satisfy.
Logical Laws
- Idempotent Law: p ∧ p = p; p ∨ p = p.
- Commutative Law: p ∧ q = q ∧ p; p ∨ q = q ∨ p.
- Associative Law: p ∧ (q ∧ r) = (p ∧ q) ∧ r; p ∨ (q ∨ r) = (p ∨ q) ∨ r.
- Distributive Law: p ∧ (q ∨ r) = (p ∧ q) ∨ (p ∧ r); p ∨ (q ∧ r) = (p ∨ q) ∧ (p ∨ r).
- De Morgan's Law: ¬(p ∧ q) = ¬p ∨ ¬q; ¬(p ∨ q) = ¬p ∧ ¬q.
- Identity Law: p ∧ T = p; p ∨ F = p.
- Complement Law: p ∧ ¬p = F; p ∨ ¬p = T.
- Absorption Law: p ∨ (p ∧ q) = p; p ∧ (p ∨ q) = p.
- Conditional Law: p → q = ¬p ∨ q.
- Biconditional Law: p ↔ q = (p → q) ∧ (q → p).
Truth Value Examples
- Evaluating truth value of statements:
- Statement: ∀x ∈ R, x² is positive.
- Truth value: False (since the square of 0 is not positive).
- Statement: ∀x ∈ R, x² is not positive.
- Truth value: True.
- Statement: Every square is a rectangle.
- Truth value: True.
- Statement: Some parallelograms are rectangles.
- Truth value: True.
- Statement: ∀x ∈ R, x² is positive.
Negations of Statements
- Negation of statement ∀x ∈ R, x² is positive: ∃x ∈ R, x² is not positive.
- Negation of statement ∀x ∈ R, x² is not positive: ∃x ∈ R, x² is positive.
- Negation of statement: Every square is a rectangle: There exists a square that is not a rectangle.
- Negation of statement: No parallelogram is a rectangle: All parallelograms are rectangles.
Proof Techniques
- Proving logical equivalences without truth tables involves algebraic transformations.
- Example renaming and manipulating expressions can demonstrate equivalency between LHS and RHS.
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Description
Test your understanding of logical implications, including converses, inverses, and contrapositives. Additionally, explore quantifiers and quantified statements related to natural numbers. This quiz is designed to deepen your grasp of these foundational concepts in logic.
