Projection is Surjection/General Version
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Theorem
For all non-empty sets $S_1, S_2, \ldots, S_j, \ldots, S_n$, the $j$th projection $\pr_j$ on $\ds \prod_{i \mathop = 1}^n S_i$ is a surjection.
Proof
Consider the $j$th projection.
As long as none of $S_1, S_2, \ldots, S_n$ is the empty set, then:
- $\ds \forall x \in S_j: \exists \tuple {s_1, s_2, \ldots, s_{j - 1}, x, s_{j + 1}, \ldots, s_n} \in \prod_{k \mathop = 1}^n S_k: \map {\pr_j} {\tuple {s_1, s_2, \ldots, s_{j - 1}, x, s_{j + 1}, \ldots, s_n} } = x$
Hence the result.
$\blacksquare$