Set Less than Cardinal Product
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Theorem
Let $S$ and $T$ be sets.
Let $T$ be nonempty.
Suppose that $S \times T \sim \card {S \times T}$.
Then:
- $\card S \le \card {S \times T}$
Proof
Let $y \in T$.
Define the mapping $f : S \to S \times T$ as follows:
- $\map f x = \tuple {x, y}$
If $\map f {x_1} = \map f {x_2}$, then $\tuple {x_1, y} = \tuple {x_2, y}$ by the definition of $f$.
It follows that $x_1 = x_2$ by Equality of Ordered Pairs.
Thus, $f: S \to S \times T$ is an injection.
By Injection implies Cardinal Inequality, it follows that $\card S \le \card {S \times T}$
$\blacksquare$
Sources
- 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 10.25$