Let $X=\{1,2,3,4,5\}$. The number of different ordered pairs $(Y,Z)$ that can be formed such that $Y\subseteq X,Z\subseteq X$ and $Y\cap Z$ is empty, is:

For each element $a\in X$, there are three mutually exclusive possibilities for it to satisfy the condition $Y\cap Z=\varphi$:$a\in Y$ and $a\notin Z$$a\notin Y$ and $a\in Z$$a\notin Y$ and $a\notin Z$ (The case $a\in Y$ and $a\in Z$ is forbidden as $Y\cap Z=\varphi$).Since there are 5 elements in $X$, and for each element there are 3 choices, the total number of ordered pairs $(Y,Z)$ is:$3\times 3\times 3\times 3\times 3=3^5$.