Thread: Set such that 1 = 0

Hi, I've been asked to find a set such that all other field axioms hold except for that 1 = 0 (multiplicative neutral = additive neutral). Currently, I reckon that the set {0} satisfies this condition, but I've also been asked to show that set is the only such example - how can I do this?

Any help is much appreciated! Thanks

Originally Posted by tunaaa

Hi, I've been asked to find a set such that all other field axioms hold except for that 1 = 0 (multiplicative neutral = additive neutral). Currently, I reckon that the set {0} satisfies this condition, but I've also been asked to show that set is the only such example - how can I do this?

Any help is much appreciated! Thanks

What other field axioms enable you to say that all elements of the field are zero?

Well, above 0 = 1, the set simply needs to satisfy:

A1 Additive commutativity
A2 Additive associativity
A3 Existence of additive neutral in set
A4 Existence of additive inverse for each element in set

M1 Multiplicative commutativity
M2 Multiplicative associativity
M3 Existence of multiplicative neutral in set
M4 Existence of multiplicative inverse for each element in set

D Distributivity of multiplication over addition.

Thanks

Originally Posted by tunaaa

Well, above 0 = 1, the set simply needs to satisfy:

A1 Additive commutativity
A2 Additive associativity
A3 Existence of additive neutral in set
A4 Existence of additive inverse for each element in set

M1 Multiplicative commutativity
M2 Multiplicative associativity
M3 Existence of multiplicative neutral in set
M4 Existence of multiplicative inverse for each element in set

D Distributivity of multiplication over addition.

Thanks

Ok...so use them. Which in particular look useful?

suppose for some integer k such that Z(k)={0,1,.......,(k-1)}
if k-1=0, then k=1 as we required Z(1)={0}.
suppose there are some other integer set such that 0 is the only element, then k-1 doesn't not equal to 0, which means k doesn't equal to 1. But if k doesn't equal to one, then k-1>0 so k>1. so k-1 must be at least 1, then there are two or more elements in this set. Therefore it is a contradiction.

so Z(1)={0} is the only case.

Why are you assuming that this is "Z modulo" some integer? 0 and 1 make sense in any ring with identity, not necessarily just integers modulo a specific integer.

For any ring with identity, z*1= z for any z. Also, you should be able to prove, by looking at z*(y+ 0)= z*y+ z*0, that z*0= 0 for all z. Now, put those together.

Thanks, but how does distributivity help me? Thanks

Originally Posted by tunaaa

Thanks, but how does distributivity help me? Thanks

You got to be kidding me.