A Fuzzy Theorem or two

An interval number A consists of a closed, bounded subset of the real numbers denoted here as A=[a_l, a_u] where a_l denotes the lower bound for the interval number A, and a_u denotes the upper bound for the interval number A. Define A as A={x: a_l<=x<=a_u}=[a_l, a_u]. Define addition of interval numbers as A+B=[a_l, a_u]+[b_l, b_u]=[a_l+b_l, a_u+b_u] Define the image of an interval number -A=-[a_l, -a_u] as [-a_u, -a_l], e. g. -[1, 2]=[-2, -1], -[-.1, .2]=[-.2, .1]. Define {0}=[0, 0] as the neutral for interval numbers, in the sense that for any interval number A, A+{0}={0}+A=A. Consider the subset N of interval numbers which lie close to {0} in the sense that n_l lies close to 0 and n_u lies close to 0.

Fuzzy Theorem: For interval numbers N which lie close to 0 in the sense that both n_l and n_u lie close to 0, if A+B approximately equals A+C, then B more or less approximately equals C.

First we'll need some lemmas.

Lemma 1: For all interval numbers A+B=B+A

Proof: A+B=[a_l, a_u]+[b_l+b_u]=[a_l+b_l, a_u+b_u]=[b_l+a_l, b_u+a_u]=[b_l, b_u]+[b_l+a_u]=B+A.

Lemma 2: For all interval numbers A+(B+C)=(A+B)+C

Proof: A+(B+C)=[a_l, a_u]+([b_l, b_u]+[c_l, c_u])=[a_l+a_u]+([b_l+c_l, b_u+c_u])
=[a_l+b_l_c_l,a_u+b_u+c_u]=[(a_l+b_l)+c_l, (a_u+b_u)+c_u]=([a_l+b_l, a_u+b_u])+[c_l, c_u]
=([a_l, a_u]+[b_l+b_u])+[c_l, c_u]=(A+B)+C

Fuzzy Lemma 3: N+(-N)=(-N)+N approximately lies close to {0}, or to put it another way, N+(-N)=(-N)+N=*{0} where =* indicates "approximately equals".

Demonstration: By lemma 1 we have N+(-N)=-N+N. N lies close to {0} in the sense that n_l and n_u lie close to 0. Consequently, -n_u and -n_l lies close to 0 also. N+(-N)=[n_l-n_u, n_u-n_l], which then approximately lies close to 0. Notice the addition of the hedge 'approxmiately' since B=[-.01, .01] may get said to lie close to {0}, while B+(-B)=[-.02, .02] has numbers a bit farther away from 0 than the numbers of [-.01, .01] and thus [-.01, .01] no longer merely lies close to {0}, but rather lies *approximately* close to {0}. But, consider C=[.01, .1]. [.01, .1]-[.01, .1]=[.01-.1, .1-.01]=[-.99, .99] which has some members which lie closer to 0 than that of [.01, .1], but also has some members which lie farther away. In this sense also, C+(-C) lies *approximately* close to {0}.

We may now proceed to demonstrate the fuzzy theorem:

We have that A+B approximately equals A+C, which I'll denote as A+B=*A+C. I'll denote more or less approximately as =**. We can then add -A to both sides to yield -A+(A+B)=*-A+(A+C). By lemma 2 we can rewrite this equation as (-A+A)+B=*(-A+A)+C. By fuzzy lemma 3 we have that (-A+A)+B=*{0}+B=B and (-A+A)+C=*{0}+C=C. So, we can more or less rewrite (-A+A)+B=*(-A+A)+C as B=*C, in other words, we have B=**C from the above.

One could also develop a similar fuzzy theorem for multiplication on intervals near {1}. Taken together these fuzzy theorems suggest an algebra which we may call an *approximate* commutative group (X, @) with the following axioms:
x_1@x_2=x_2@x_1
(x_1@x_2)@x_3=x_1@(x_2@x_3)
x_1@{0}={0}@x_1=x_1
x_1@(-x_1)=(-x_1)@(x_1)=*{0}