Prove: every set with an infinite subset is infinite

**noles2188** · March 31st 2009, 02:05 PM

Prove that every set that has an infinite subset is infinite.
Also, prove that every subset of a finite set is finite.

**Plato** · March 31st 2009, 02:12 PM

Originally Posted by noles2188

Prove that every set that has an infinite subset is infinite.
Also, prove that every subset of a finite set is finite.

The proof completely depends on how you textbook defines infinite.
What definition does it use?

**noles2188** · March 31st 2009, 03:07 PM

Here is the definition from the text: "The statement that the set A is infinite means that there is a nonempty proper subset B of A such that there is a one-to-one correspondence between A and B; A is finite means that A is not infinite."

**Plato** · March 31st 2009, 04:22 PM

Suppose that $B$ is an infinite and $B \subseteq A$ we want to show that $A$ is infinite.
By the definition $\left( {\exists C} \right)\left[ {C \varsubsetneq B} \right]$ and there is a bijection such that $f:B \mapsto C$ .
It should be clear that $D = \left( {A\backslash B} \right) \cup C$ is a proper subset of $A$ .

Define a function $g:A \mapsto D$ as $g(x) = \left\{ {\begin{array}{rl} {f(x),} & {x \in B} \\ {x,} & {x \in A\backslash B} \\ \end{array} } \right.$
Now your task is to show that $g$ is a bijection.

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