If $\mathrm{P}=\{1{,}2\}$, form the set $\mathrm{P} \times \mathrm{P} \times \mathrm{P}$.

Step 1: Understand the Cartesian Product

The Cartesian product of two sets $A$ and $B$, denoted as $A \times B$, is the set of all possible ordered pairs $(a,b)$ where $a$ is an element of $A$ and $b$ is an element of $B$. For three sets, $A \times B \times C$ is the set of all ordered triplets $(a,b,c)$ where $a \in A$, $b \in B$, and $c \in C$.

Step 2: Identify the given set

We are given the set $\mathrm{P}$ which contains two elements, $1$ and $2$. We need to find the Cartesian product of $\mathrm{P}$ with itself three times, i.e., $\mathrm{P} \times \mathrm{P} \times \mathrm{P}$.

Step 3: Calculate P × P

First, let's find the Cartesian product of $\mathrm{P}$ with itself. This involves forming all possible ordered pairs where both elements come from $\mathrm{P}$. The elements are $(1{,}1)$, $(1{,}2)$, $(2{,}1)$, and $(2{,}2)$.

Step 4: Calculate (P × P) × P

Now, we need to find the Cartesian product of $(\mathrm{P} \times \mathrm{P})$ with $\mathrm{P}$. This means we will take each ordered pair from $(\mathrm{P} \times \mathrm{P})$ and combine it with each element from $\mathrm{P}$ to form ordered triplets.

Step 5: List all elements of P × P × P

By combining each pair from $\mathrm{P} \times \mathrm{P}$ with each element from $\mathrm{P}$, we get the following ordered triplets: $(1{,}1{,}1)$, $(1{,}1{,}2)$, $(1{,}2{,}1)$, $(1{,}2{,}2)$, $(2{,}1{,}1)$, $(2{,}1{,}2)$, $(2{,}2{,}1)$, and $(2{,}2{,}2)$. There are $2 \times 2 \times 2 = 8$ elements in total.

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