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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.