## 2 Background and Related Work
### JAC based on Interrelated Attributes  
In this subsection, we employ JAC measure (Equation (2)) to introduce the application of the interrelated attributes between two items and the number of interrelated attributes that each item owns. Note that the interrelated attributes can also be applied to other approaches which utilize item attributes in generating recommendations. Let \(AT\) be the set of attributes that comprises \(l\) attributes and \(i_{x},i_{y}\in I\) be any two items. Suppose item \(i_{x}\) owns \(m_{x}\) attributes and item \(i_{y}\) owns \(m_{y}\) attributes, respectively. By changing the order of attributes freely with respect to Lemma 1, without loss of generality, the following condition can be assumed:* Both items \(i_{x}\) and \(i_{y}\) have \(n_{1}(n_{1}\geq 0)\) attributes \(a_{1},\ldots,\ a_{n_{1}}\). * Item \(i_{x}\) also has all \(n_{2}(n_{2}\geq 0)\) attributes \(b_{1},\ldots,\ b_{n_{2}}\) but item \(i_{y}\) does not have any of these attributes. * Item \(i_{y}\) also has all \(n_{3}(n_{3}\geq 0)\) attributes \(c_{1},\ldots,\ c_{n_{3}}\) but item \(i_{x}\) does not have any of these attributes. * Both items \(i_{x}\) and \(i_{y}\) do not have \(n_{4}(n_{4}\geq 0)\) attributes \(d_{1},\ldots,\ d_{n_{4}}\),  
where, the conditions of the number of attributes that item \(i_{x}\) and \(i_{y}\) own, \(n_{1}+n_{2}+n_{3}+n_{4}=l\), \(n_{1}+n_{2}=\ m_{x}\), and \(n_{1}+n_{3}=\ m_{y}\) hold, respectively. Using Theorem 1, the number of interrelated attributes that item \(i_{x}\) owns can be obtained as follows:  
\[ \begin{array}{ll}num(x;l,m_{x})&=C_{n_{x}}^{2}+\ m_{x}(l-m_{x})\\ &=C_{n_{1}+n_{2}}^{2}+(n_{1}+n_{2})(n_{3}+n_{4})\\ &=C_{n_{1}+n_{2}}^{2}+n_{1}n_{4}+n_{1}n_{3}+n_{2}n_{3}+n_{2}n_{4}\end{array}  \tag{12}\]  
Similarly, the number of interrelated attributes that item \(i_{y}\) owns is  
\[num\Big{(}y;l,m_{y}\Big{)}=C_{n_{1}+n_{3}}^{2}+n_{1}n_{4}+n_{1}n_{2}+n_{2}n_{3} +n_{3}n_{4} \tag{13}\]  
Because both items \(i_{x}\) and \(i_{y}\) have \(n_{1}(n_{1}\geq 0)\) attributes \(a_{1},\ldots,\ a_{n_{1}}\), either there is no interrelated attribute \(a_{i}=_{eq}a_{j}(1\leq i<j\leq n_{1})\) when \(n_{1}=0\) or \(n_{1}=1\), or there is \(C_{n_{1}}^{2}\) attributes \(a_{i}=_{eq}a_{j}(n_{1}\geq 2\) and \(1\leq i<j\leq n_{1})\) that both items \(i_{x}\) and \(i_{y}\) own commonly. Hence, in the case of item \(i_{x}\), the rest \(C_{n_{1}+n_{2}}^{2}-C_{n_{1}}^{2}\) interrelated attributes based on the relation \(=_{eq}\) are owned by only item \(i_{x}\) and not item \(i_{y}\). Similarly, in the case of item \(i_{y}\), \(C_{n_{1}+n_{3}}^{2}-C_{n_{1}}^{2}\) interrelated attributes based on the relation \(=_{eq}\) are owned by only item \(i_{y}\) and not item \(i_{x}\). Moreover, similar to the above discussion, both item \(i_{x}\) and item \(i_{y}\) do not have \(n_{4}(n_{4}\geq 0)\) attributes \(d_{1},\ldots,\ d_{n_{4}}\), either because there is no interrelated attribute \(a_{i}>_{mo}d_{q}(1\leq i\leq n_{1}\) and \(1\leq q\leq n_{4})\) if when \(n_{1}=\ 0\) or \(n_{4}=\ 0\) or \(n_{1}n_{4}\) interrelated attributes \(a_{i}>_{mo}d_{q}\) that item \(i_{x}\) and item \(i_{y}\) own commonly. Based on the structure of attributes that item \(i_{x}\) and item \(i_{y}\) own, it is obvious that the remaining interrelated attributes denoted by \(>_{mo}(\) or \(<_{le})\) are owned by either item \(i_{x}\) or item \(i_{y}\) only. These discussions conclude the following results about the relationship between JAC similarity between item \(i_{x}\) and item \(i_{y}\) and the number of interrelated attributes that item \(i_{x}\) and item \(i_{y}\) own commonly. Theorem 2: _Let AT be a set of attributes that comprises \(l\) attributes, item \(i_{x},i_{y}\in l\) be any two items and suppose that item \(i_{x}\) owns \(m_{x}\) attributes and item \(i_{y}\) owns \(m_{y}\) attributes, respectively.