Consider the following axiom system: Undefined terms: point, line, incident with Defined terms: Two lines are said to be parallel if and only if no point incident with both of them. AXIOMS: Axiom 1. Any two distinct points determine a unique line Axiom 2. Any line is incident with at least two points. Axiom3. There is exactly 4 distinct points. Axiom 4. No line incident with all points. i. Is the following statement independent from this Axiom System? Explain why? S: No parallel lines exists. Is this axiom system categorical? Explain. Is it a complete axiom system? Explain. ii. iii.

Consider the following axiom system: Undefined terms: point, line, incident with Defined terms: Two lines are said to be parallel if and only if no point incident with both of them. AXIOMS: Axiom 1. Any two distinct points determine a unique line Axiom 2. Any line is incident with at least two points. Axiom3. There is exactly 4 distinct points. Axiom 4. No line incident with all points. i. Is the following statement independent from this Axiom System? Explain why? S: No parallel lines exists. Is this axiom system categorical? Explain. Is it a complete axiom system? Explain. ii. iii.

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Description: Consider the following axiom system:Undefined terms: point, line, incident withDefined terms: Two lines are said to be parallel if and only if no pointincident with both of them.ΑXΙΟMS:Axiom 1. Any two distinct points determine a unique lineAxiom 2.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.

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Consider the following axiom system:
Undefined terms: point, line, incident with
Defined terms: Two lines are said to be parallel if and only if no point
incident with both of them.
ΑXΙΟMS:
Axiom 1. Any two distinct points determine a unique line
Axiom 2. Any line is incident with at least two points.
Axiom3. There is exactly 4 distinct points.
Axiom 4. No line incident with all points.
Is the following statement independent from this Axiom System?
Explain why?
i.
S: No parallel lines exists.
ii.
Is this axiom system categorical? Explain.
Is it a complete axiom system? Explain.
iii.

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