Infinite Set is Equivalent to Proper Subset/Examples/Even Integers
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Examples of Use of Infinite Set is Equivalent to Proper Subset
Let $\Z$ be the set of integers.
By Integers are Countably Infinite, $\Z$ is infinite.
Let $E$ be the set of all even integers.
We have that, for example, $3 \in \Z$ but $3 \notin E$
Hence $E$ is a proper subset of $\Z$
Let $f: \Z \to E$ be the mapping defined as:
- $\forall x \in \Z: \map f x = 2 x$
Then $f$ is a bijection.
Hence a fortiori $\Z$ and $E$ are equivalent.
Sources
- 1979: John E. Hopcroft and Jeffrey D. Ullman: Introduction to Automata Theory, Languages, and Computation ... (previous) ... (next): Chapter $1$: Preliminaries: $1.4$ Set Notation: Infinite sets