Finite Axiomatic Geometry
Not what you're looking for?
Geometry
Finite Axiomatic Geometry
Finite Axiomatic Geometry
This abstract geometry confuses me and I just can't follow this logic. Here is a set up of the question. Please help with some insight.
Consider the following axiom system.
The undefined terms are point, line, and on.
Definition: two lines are parallel if there is no point on both of them.
Axioms:
I. Given any two distinct points, there exactly one line on both of them.
II. Every line has at least one point on it.
III. Given any line, there is at least one other line that is parallel to it.
IV. There exists at least one line.
[The book defines each axiom an "undefined term" called a "interpretation of a system."]
[If, for a given interpretation of a system, all the axioms are "correct" statements, we call the interpretation a "MODEL"]
Questions:
(a) Show that this axiom system is consistent by finding a model for it.
(b) Using models, show that Axiom III is independent of the other axioms.
(c) Show that this axiom system is not complete.
Purchase this Solution
Solution Summary
This solution is comprised of a detailed explanation of the Finite Axiomatic Geometry.
It contains step-by-step explanation for the following problem:
Finite Axiomatic Geometry
This abstract geometry confuses me and I just can't follow this logic. Here is a set up of the question. Please help with some insight.
Consider the following axiom system.
The undefined terms are point, line, and on.
Definition: two lines are parallel if there is no point on both of them.
Axioms:
I. Given any two distinct points, there exactly one line on both of them.
II. Every line has at least one point on it.
III. Given any line, there is at least one other line that is parallel to it.
IV. There exists at least one line.
[The book defines each axiom an "undefined term" called a "interpretation of a system."]
[If, for a given interpretation of a system, all the axioms are "correct" statements, we call the interpretation a "MODEL"]
Questions:
(a) Show that this axiom system is consistent by finding a model for it.
(b) Using models, show that Axiom III is independent of the other axioms.
(c) Show that this axiom system is not complete.
Solution contains detailed step-by-step explanation.
Solution Preview
Geometry
Finite Axiomatic Geometry
Finite Axiomatic Geometry
This abstract geometry confuses me and I just can't follow this logic. Here is a set up of the question. Please help with some insight.
Consider the following ...
Education
- BSc, Manipur University
- MSc, Kanpur University
Recent Feedback
- "Thanks this really helped."
- "Sorry for the delay, I was unable to be online during the holiday. The post is very helpful."
- "Very nice thank you"
- "Thank you a million!!! Would happen to understand any of the other tensor problems i have posted???"
- "You are awesome. Thank you"
Purchase this Solution
Free BrainMass Quizzes
Know Your Linear Equations
Each question is a choice-summary multiple choice question that will present you with a linear equation and then make 4 statements about that equation. You must determine which of the 4 statements are true (if any) in regards to the equation.
Probability Quiz
Some questions on probability
Graphs and Functions
This quiz helps you easily identify a function and test your understanding of ranges, domains , function inverses and transformations.
Geometry - Real Life Application Problems
Understanding of how geometry applies to in real-world contexts
Exponential Expressions
In this quiz, you will have a chance to practice basic terminology of exponential expressions and how to evaluate them.