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This is a question about axioms in my Geometry class. There are two types of axioms that are known in the picture: field axioms and order axioms. Which of the axioms apply to the set of whole numbers? Which do not?

This is a question about axioms in my Geometry class. There are two types of axioms that are known in the picture: field axioms and order axioms. Which of the axioms apply to the set of whole numbers? Which do not?

Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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This is a question about axioms in my Geometry class.

There are two types of axioms that are known in the picture: field axioms and order axioms. Which of the axioms apply to the set of whole numbers? Which do not?

We will start with the field axioms. We know that the set of real numbers is a nonempty set. To make it a field, we observe that the usual notion of addition and multiplication on the set of real numbers satisfies the following properties. Let XX.z be real numbers. 1. Closure (Addition) – xty, is a real number. 2. Closure (Multiplication) – x*y is a real number. 3. Associativity (Addition) – (xtx)+z=x+(xtz) 4. Associativity (Multiplication) – (x*y)*z=x*(y*z) 5. Identity (Addition) – There exists a real number 0 called the additive identity, such that x+0=x no matter what x is. 6. Identity (Multiplication) – There exists a real number 1 called the multiplicative identity, such that x*1=x no matter what x is. 7. Inverse (Addition) – For each x, there corresponds a real number -x called the additive inverse, such that x+(-x)=0. 8. Inverse (Multiplication) – For each nonzero x, there corresponds a real number 1/x called the multiplicative inverse, such that x*(1/x)=1. 9. Distributivity – Multiplication is distributive over addition, that is, x*(xtz)=x*xtxz Next we have the order axioms. Simply put, these allow us to arrange the real numbers in a line. We observe that the usual notion of "less than" satisfies the following properties. Let xV.z be real numbers. 1. Trichotomy – Exactly one of the following is true: x<y, x=y or y<x (also written x>y). 2. Transitivity – If x<y and y<z then x<z. 3. Addition axiom of order – If x<y then xtz<xtz. 4. Multiplicative axiom of order – If x<y and z>0 then x*z<y*z.
Transcribed Image Text:We will start with the field axioms. We know that the set of real numbers is a nonempty set. To make it a field, we observe that the usual notion of addition and multiplication on the set of real numbers satisfies the following properties. Let XX.z be real numbers. 1. Closure (Addition) – xty, is a real number. 2. Closure (Multiplication) – x*y is a real number. 3. Associativity (Addition) – (xtx)+z=x+(xtz) 4. Associativity (Multiplication) – (x*y)*z=x*(y*z) 5. Identity (Addition) – There exists a real number 0 called the additive identity, such that x+0=x no matter what x is. 6. Identity (Multiplication) – There exists a real number 1 called the multiplicative identity, such that x*1=x no matter what x is. 7. Inverse (Addition) – For each x, there corresponds a real number -x called the additive inverse, such that x+(-x)=0. 8. Inverse (Multiplication) – For each nonzero x, there corresponds a real number 1/x called the multiplicative inverse, such that x*(1/x)=1. 9. Distributivity – Multiplication is distributive over addition, that is, x*(xtz)=x*xtxz Next we have the order axioms. Simply put, these allow us to arrange the real numbers in a line. We observe that the usual notion of "less than" satisfies the following properties. Let xV.z be real numbers. 1. Trichotomy – Exactly one of the following is true: x<y, x=y or y<x (also written x>y). 2. Transitivity – If x<y and y<z then x<z. 3. Addition axiom of order – If x<y then xtz<xtz. 4. Multiplicative axiom of order – If x<y and z>0 then x*z<y*z.
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