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.AXIOMS:Axiom 1. Any two distinct points determine a unique lineAxiom 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: For any line t and any point P not on e, there is at least two lines incidentwith P and parallel to lIs this axiom system categorical? Explain.Is it a complete axiom system? Explain.ii.iii.iv.Give an example of a non-consistent axiom system, other than the onegiven in the lecture.

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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. AXIΟMS: 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.

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. AXIΟMS: 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.

Advanced Engineering Mathematics

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ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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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.
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.
Is the following statement independent from this Axiom System?
Explain why?
i.
S: For any line t and any point P not on e, there is at least two lines incident
with P and parallel to l
Is this axiom system categorical? Explain.
Is it a complete axiom system? Explain.
ii.
iii.
iv.
Give an example of a non-consistent axiom system, other than the one
given in the lecture.

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