# Mapping from Set to Ordinate of Cartesian Product is Injection

## Theorem

Let $S$ and $T$ be sets such that $T \ne \O$.

Let $S \times T$ denote their cartesian product.

Let $t \in T$ be given.

Let $j_t \subseteq S \times \paren {S \times T}$ be the mapping from $S$ to $S \times T$ defined as:

$\forall s \in \S: \map {j_t} s = \tuple {s, t}$

Then $j_t$ is an injection.

## Proof

It has been shown in Correspondence between Set and Ordinate of Cartesian Product is Mapping that $j_t$ is a mapping.

Now it is to be shown that $j_t$ is injective, that is:

$\forall s_1, s_2 \in S: \map {j_t} {s_1} = \map {j_t} {s_2} \implies s_1 = s_2$

We have that:

 $\displaystyle \map {j_t} {s_1}$ $=$ $\displaystyle \map {j_t} {s_2}$ $\displaystyle \leadsto \ \$ $\displaystyle \tuple {s_1, t}$ $=$ $\displaystyle \tuple {s_2, t}$ Definition of $j_t$ $\displaystyle \leadsto \ \$ $\displaystyle s_1$ $=$ $\displaystyle s_2$ Definition of Ordered Pair

Hence the result.

$\blacksquare$