- Definition.$1$
$$X\times_1 Y=\{ (x,y)=\{\{x\},\{x,y\}\}: x\in X, y\in Y\}$$ - Definition.$2$
$$X\times_2 Y=\left\{ f\in Fonk\left( \left\{ 0,1\right\},X\cup Y\right) : f(0)\in X, f(1)\in Y\right\}$$
Show there is a bijection between Definition $1$ of Cartesian Product and Definition $2$ of Cartesian Product.
My proof trying:
Let $T$ be a function from $X\times_1 Y$ to $X\times_2 Y$. We will show that $T$ is bijection.
Case 1:
One-to-one. Let $a_1,a_2$ be any elements of domain of $T$. Then, $a_1$ of the form is $a_1:=(x_1,y_1)$, and $a_2$ of the form is $a_2:=(x_2,y_2)$. We need to show $T(a_1)=T(a_2)$.
I couldn't continue my proof, can you help?
We can define $$\begin{align}T\colon X\times_2 Y&\to X\times_1 Y\\f&\mapsto(f(0),f(1))\end{align}$$ (because $f\in X\times_2 Y$ implies $f(0)\in X$ and $f(1)\in Y$ as required).
Showing that $T$ is injective and surjective is straightforward. Alternatively, also define $$\begin{align}U\colon X\times_1 Y&\to X\times_2 Y\\(x,y)&\mapsto t\mapsto \begin{cases}x&t=0\\y&t=1\end{cases}\end{align}$$ and show that $U\circ T$ and $T\circ U$ are the identities on the two versions of product.