## 2 The Norm Systems
### The Repeatability Norms
A sequence of logarithmic functions \(Sc_{n}(x)=1-\log_{(1-1/n)}x\), for \(n>1,x\in(0,1]\), when \(Sc_{1}(x)=x\) and \(Sc_{n}(0)=0\), is a sequence of scaling functions, since:  
1. \(Sc_{1}(x)=x\) and functions \(Sc_{n}\) are strictly growing,
2. \(Sc_{n}(0)=0,Sc_{n}(1)=1-\log_{(1-1/n)}1=1\), for any \(n>1\),
3. For any \(x\in(0,1]\), \(x\leq 1\) iff \(1/2\leq 1/2x\) iff \(\log_{1/2}1/2x\leq\log_{1/2}1/2\) iff \(1-\log_{1/2}x\leq 1\) iff \(Sc_{2}(x)\leq Sc_{1}(x)\),  
for any \(n>1,x\in(0,1]\), \(1-1/n\leq 1-1/(n+1)\) iff \(\log_{(1-1/n)}x\leq\log_{(1-1/(n+1))}x\) iff \(1-\log_{(1-1/(n+1))}x\leq 1-\log_{(1-1/n)}x\) iff \(Sc_{n+1}(x)\leq Sc_{n}(x)\),
4. For \(u,v\in[0,1]\), when \(uv=0\) or for \(uv=1\) there is \(0\leq Sc_{1}(u)+Sc_{1}(v)\leq 1\),  
for \(uv<0\) there is such \(k>1\), that \(uv\leq 1-1/k\) iff \(\log_{(1-1/k)}(1-1/k)\leq\log_{(1-1/k)}uv\) iff \(2-\log_{(1-1/k)}uv\leq 1\) iff \((1-\log_{(1-1/k)}u)+(1-\log_{(1-1/k)}v)\) iff \(0\leq Sc_{k}(u)+Sc_{k}(v)\leq 1\),
5. \(Sc_{1}^{-1}(x)=x\) and for \(n>1\), \(Sc_{n}^{-1}(x)=(1-1/n)^{x}\). Example 5: The power scale \(A\) sequence of power functions \(Sc_{n}(x)=x^{n}\), for \(n>0,x\in[0,1]\) is a sequence of scaling functions, since:  
1. \(Sc_{1}(x)=x\) and functions \(Sc_{n}\) are strictly growing,
2. \(Sc_{n}(0)=0,Sc_{n}(1)=1\), for any \(n\),
3. \(x\leq 1\) iff \(x^{(n+1)}\leq x^{n}\) iff \(Sc_{n+1}(x)\leq Sc_{n}(x)\), for any \(n\) and \(x\in[0,1]\),_._
4. _For_ \(u,v\in[0,1]\) _exists such_ \(k1,k2\)_, that_ \(u^{k1}\leq 1/2\) _and_ \(v^{k2}\leq 1/2\)_, and when_ \(k=\max\{k1,k2\}\)_, then_ \(0\leq u^{k}+v^{k}\leq 1\)_, hence_ \(0\leq Sc_{k}(u)+Sc_{k}(v)\leq 1\)_,_
5. _For_ \(n>0\)_,_ \(Sc_{n}^{-1}(x)=x^{1/n}\)_._  
**Definition 5**. : _Let \(F=\{f:X\rightarrow[0,1]\}\) be the set of all fuzzy sets on non-empty space \(X\)._  
_Any function_ \(p:X\rightarrow[0,1]\times[0,1]\)_, for any_ \(\mu_{p},v_{p}\in F\) _and some natural number_ \(n>0\)_, is defined by formulas:_  
\(\mu_{p}=\{\langle x,u\rangle:\)_exists_ \(v\in[0,1]\) _such that_ \(p(x)=\langle u,v\rangle,x\in X\}\)_,_  
\(v_{p}=\{\langle x,v\rangle:\)_exists_ \(u\in[0,1]\) _such that_ \(p(x)=\langle u,v\rangle,x\in X\}\)_,_  
\(p=\{\langle x,\langle\mu_{p}(x),v_{p}(x)\rangle\rangle:x\in X\}\)_,_  
_will be called the **n-scaling intuitionistic fuzzy sets** (\(n\)-ScIFS) for sequences \(\{Scn\},\{Scn^{\prime}\}\) of scaling functions, if the condition is met:_  
\(0\leq Sc_{n}(\mu_{p}(x))+Sc_{n}^{\prime}(v_{p}(x))\leq 1\)_, for any_ \(x\in X\)_._  
_The \(n\)-ScIFS is a set of all \(n\)-scaling intuitionistic fuzzy sets for sequences \(\{Scn\},\{Scn^{\prime}\}\) of scaling functions._  
The fuzzy set \(\mu_{p}\) indicates the **membership** and \(v_{p}\) indicates the **non-membership**. Let \(p=_{df}\langle\mu_{p},v_{p}\rangle\) represent the n-fuzzy number \(n\)-ScIFN. Then \(p(x)=_{df}\langle\mu_{p}(x),v_{p}(x)\rangle\) and \(\langle\mu_{p},v_{p}\rangle(x)=_{df}\langle\mu_{p}(x),v_{p}(x)\rangle\). For this reason, all operations and relations on the \(n\)-ScIFN will still be considered as defined on the set:  
\(\textbf{n-ScIFN}=_{df}\{\langle u,v\rangle\in[0,1]\times[0,1]:0\leq Sc_{n}( \mu)+Sc_{n}(v)\leq 1\}\). Moreover, there is:  
**Fact 2**. : \(\textbf{n-ScIFN}=\{p(x):x\in X,p\in n\text{-ScIFS}\}\)_,_  
_Any function_ \(p:X\rightarrow\textbf{n-ScIFN}\) _is the n-scaling intuitionistic fuzzy sets for sequences \(\{Scn\},\{Scn^{\prime}\}\) of scaling functions._  
When \(n=1\), then _1-ScIFS_s are the intuitionistic fuzzy sets (IFS), which have been studied by [PERSON] [25], and _2-ScIFSs_ for \(Sc_{2}(x)=Sc_{2}^{\prime}(x)=x^{2}\) are the PFS of [PERSON] [26]. From Definition 4 and from a simple arithmetic property: \(0\leq Sc_{n+1}(\mu_{p}(x))+Sc_{n+1}^{\prime}(v_{p}(x))\leq Sc_{n}(\mu_{p}(x))+ Sc_{n}^{\prime}(v_{p}(x))\leq 1\) for \(\langle\mu_{p},v_{p}\rangle\in n\text{-ScIFS}\), there is:  
**Remark 2**.