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Step-by-Step Solution
Step 1: Understand the problem
We have a set $X = \{1, 2, 3, 4, 5\}$. We want to form ordered pairs of subsets $(Y, Z)$ such that $Y \subseteq X$, $Z \subseteq X$, and $Y \cap Z = \varnothing$. We need to find the total number of such possible ordered pairs.
Step 2: Analyze possible choices for each element
Consider any element $x_i$ in $X$. When creating the subsets $Y$ and $Z$, we look at how $x_i$ can be placed with respect to $Y$ and $Z$:
Choice 1: $x_i \in Y$ and $x_i \in Z$ (not allowed because $Y \cap Z \neq \varnothing$).
Choice 2: $x_i \in Y$ and $x_i \notin Z$ (allowed because $x_i$ is only in $Y$).
Choice 3: $x_i \notin Y$ and $x_i \in Z$ (allowed because $x_i$ is only in $Z$).
Choice 4: $x_i \notin Y$ and $x_i \notin Z$ (allowed because $x_i$ is in neither set).
Out of these four possible ways, only three keep $x_i$ from being in both $Y$ and $Z$ simultaneously and thus keep $Y \cap Z = \varnothing$.
Step 3: Count the total number of ways
Since each of the 5 elements of $X$ can be independently assigned to either
Y only, Z only, or neither Y nor Z, there are 3 valid choices per element. Hence, for 5 elements, the total number of ways is:
$3^5$
Step 4: Final answer
The number of different ordered pairs $(Y, Z)$ such that $Y \subseteq X$, $Z \subseteq X$, and $Y \cap Z = \varnothing$ is
$3^5.$
