Addition of Cardinals is non Decreasing - kipawaa/Proof-Tree GitHub Wiki

Statement

The addition of Cardinal Numbers is non-decreasing, i.e. if $\lambda, \kappa$ are Cardinal Numbers then $\kappa + \lambda \geq \kappa$.

Explanation

Proof(s)

Let $\kappa = \lvert A \rvert, \lambda = \lvert B \rvert$ such that $A \cap B = \emptyset$.
Then $\kappa + \lambda = \lvert A \cup B \rvert$ by the definition of Addition-of-Cardinal-Numbers.
Define the Function $f : A \to A \cup B$ such that $f(a) = a$.
Then $f$ is an Injective Function, and hence $\lvert A \rvert \leq \lvert A \cup B \rvert$ by the Ordering of Cardinal Numbers.
This gives that $\kappa \leq \kappa + \lambda$, as wanted.

History

Applications

Links

Dependencies

• Injective Function
• Function
• Cardinal Numbers
• Ordering of Cardinal Numbers

Dependents

Sources

• Hrbacek, K., & Jech, T. (1999). Introduction to Set Theory, Revised and Expanded (3rd ed.). CRC Press. https://doi.org/10.1201/9781315274096