Prove: every set with an infinite subset is infinite

• Mar 31st 2009, 02:05 PM
noles2188
Prove: every set with an infinite subset is infinite
Prove that every set that has an infinite subset is infinite.
Also, prove that every subset of a finite set is finite.
• Mar 31st 2009, 02:12 PM
Plato
Quote:

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?
• Mar 31st 2009, 03:07 PM
noles2188
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."
• Mar 31st 2009, 04:22 PM
Plato
Suppose that $\displaystyle B$ is an infinite and $\displaystyle B \subseteq A$ we want to show that $\displaystyle A$ is infinite.
By the definition $\displaystyle \left( {\exists C} \right)\left[ {C \varsubsetneq B} \right]$ and there is a bijection such that $\displaystyle f:B \mapsto C$.
It should be clear that $\displaystyle D = \left( {A\backslash B} \right) \cup C$ is a proper subset of $\displaystyle A$.

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