Ordering of Cardinals Compatible with Cardinal Sum
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Theorem
Let $\mathbf a$, $\mathbf b$ and $\mathbf c$ be cardinals.
Then:
- $\mathbf a \le \mathbf b \implies \mathbf a + \mathbf c \le \mathbf b + \mathbf c$
where $\mathbf a \mathbf c$ denotes the sum of $\mathbf a$ and $\mathbf c$.
Proof
Let $\mathbf a = \map \Card A$, $\mathbf b = \map \Card B$ and $\mathbf c = \map \Card C$ for some sets $A$, $B$ and $C$.
Let $C$ be chosen such that $A \cap C = \O = B \cap C$.
Let $\mathbf a \le \mathbf b$.
Then by definition of cardinal, there exists an injection $f: A \to B$.
Then the mapping $h: A \cup C \to B \cup C$ defined as:
- $\forall x \in A \cup C: \map h x = \begin{cases} \map f x & : x \in A \\ x & : x \in C \end{cases}$
Let $a_1 \in A \cup C$ and $a_2 \in A \cup C$ such that:
- $\map h {a_1} = \map h {a_2}$
Then:
\(\ds a_1\) | \(\in\) | \(\ds C\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds a_1\) | \(=\) | \(\ds \map h {a_1}\) | Definition of $h$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds a_1\) | \(=\) | \(\ds \map h {a_2}\) | as $\map h {a_1} = \map h {a_2}$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds a_1\) | \(=\) | \(\ds a_2\) | as $\map h {a_2}$ must also be in $C$ |
and:
\(\ds a_1\) | \(\notin\) | \(\ds C\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map h {a_1}\) | \(=\) | \(\ds \map f {a_1}\) | Definition of $h$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map h {a_2}\) | \(=\) | \(\ds \map f {a_1}\) | as $\map h {a_1} = \map h {a_2}$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map h {a_2}\) | \(=\) | \(\ds \map f {a_2}\) | as $\map h {a_2}$ must also not be in $C$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map f {a_1}\) | \(=\) | \(\ds \map f {a_2}\) | as $\map h {a_1} = \map h {a_2}$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds a_1\) | \(=\) | \(\ds a_2\) | as $f$ is an injection |
So:
- $\map h {a_1} = \map h {a_2} \implies a_1 = a_2$
demonstrating that $h$ is an injection.
So, by definition of sum of cardinals:
- $\mathbf a + \mathbf c \le \mathbf b + \mathbf c$
$\blacksquare$
Sources
- 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 8$: Theorem $8.7$