Find feild where 1 = 0 and show that it is the only one

  • Thread starter rosh300
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In summary, the statement "Find field where 1 = 0" means to find a mathematical space where the value of 1 is equal to 0. This can be true in certain abstract algebraic structures but not in most mathematical systems. "Showing that it is the only one" means to prove that this field is the only one that satisfies the condition. Finding this field has important implications in abstract algebra, mathematical logic, and other areas such as computer science and quantum mechanics. Although not considered a real field, it can have applications in representing "false" or "not possible" in programming languages.
  • #1
rosh300
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Homework Statement


Define a paddock to be a set in which A1 - A4, M1 - M4 and D holds but instead of 1[tex]\neq[/tex] 0, we have 1 = 0. Find an example of a paddock, and show that your example is the only one

Homework Equations


A1 - A4, M1 - M4 and D are all axioms

addition axioms
A1: a + b = b + a
A2: (a + b) + c = a + (b + c)
A3: there is a 0 s.t a + 0 = 0 + a = a
A4 for every a there exsist -a s.t a + (-a) = 0

multiplication axioms:
M1: a.b = b.a
M2: (a.b).c = a.(b.c)
M3: there is a 1 s.t a.1 = 1.a = a
M4: for every a there exsist a-1 s.t a.a-1 = 1

D (a+b)c = ab + bc
a, b, c, 1, 0, inverses all belong in the set

The Attempt at a Solution


the set {1, 0} closed under +1 and x1 (modulo multiplication of 1 and modulo addition of 1).
1 = 0 because both 0 and 1 are the neutral element,
if it is true how how would i go about showing its the only one. I am guessing it requires proof by contradiction.
 
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  • #2
M3 tells you that a*1= a for all a.

By D, a*(b+ 0)= a*b+ a*0 but by A3, b+ 0 = b so a*(b+ 0)= a*b. That is, a*(b+0) is equal to both a*b and a*b+ a*0 so they are equal: a*b= a*b+ a*0 and, adding the additive inverse of a*b to both sides, a*0= 0 for all a.

Now you have both a*1= a and a*0= 0 as well as 0= 1. Put them together.

("Paddock" instead of "field". That's cute.)
 
  • #3
HallsofIvy said:
M3 tells you that a*1= a for all a.

By D, a*(b+ 0)= a*b+ a*0 but by A3, b+ 0 = b so a*(b+ 0)= a*b. That is, a*(b+0) is equal to both a*b and a*b+ a*0 so they are equal: a*b= a*b+ a*0 and, adding the additive inverse of a*b to both sides, a*0= 0 for all a.

Now you have both a*1= a and a*0= 0 as well as 0= 1. Put them together.

("Paddock" instead of "field". That's cute.)

May I ask, how does putting 'a*1= a and a*0= 0 as well as 0= 1' together explicitly shows that this example of a paddock is the only one?
 
  • #4
vintwc said:
May I ask, how does putting 'a*1= a and a*0= 0 as well as 0= 1' together explicitly shows that this example of a paddock is the only one?
a*1= a and a*0= 0 are the same thing because 0= 1. Therefore a= 0 for all a. The "paddock" contains only a single element.
 

What does the statement "Find field where 1 = 0" mean?

This statement means to find a field or mathematical space where the value of 1 is equal to 0. In most mathematical systems, this is not possible as 1 is always considered to be greater than 0. However, in certain abstract algebraic structures, this statement can be true.

What does it mean to "show that it is the only one"?

This means to prove that the field found in the previous statement is the only one that satisfies the condition of 1 being equal to 0. This can be done through mathematical proofs or by showing that other possible fields do not meet the criteria.

Why is it important to find this field?

Finding this field can have important implications in abstract algebra and mathematical logic. It can help us better understand the properties and limitations of different mathematical systems and can also have applications in computer science and cryptography.

Is the field where 1 = 0 a real field?

No, in most cases, this field is not considered to be a real field as it violates one of the fundamental properties of a field, which is that 1 must be greater than 0. However, it can exist in abstract algebraic structures such as the Boolean algebra.

How is this statement relevant to other areas of science?

Although this statement is primarily used in mathematics, it can also have applications in other areas of science. For example, in computer science, this statement can be used in programming languages to represent the concept of "false" or "not possible". It may also have implications in quantum mechanics and other theoretical physics concepts.

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