How to prove that if $X$ is a subset of $Y$ , and $X$ is inﬁnite, then $Y$ is inﬁnite

Question

How do I prove this?

Prove that if $X$ is a subset of $Y$ , and $X$ is inﬁnite, then $Y$ is inﬁnite.

Thanks in advance.

As you can see from one of the answers, you need to specify how you have had infinite defined. Note that the contrapositive of the statement is that any subset of a finite set is finite. — Tobias Kildetoft, Oct 02 '13 at 12:17

score 4 · Accepted Answer · answered Oct 02 '13 at 12:15

It depends on how you define infinite set.

The definition that I have been familiar with is that a set $X$ is infinite if there exists a proper subset $S$ such that their is a bijective correspondence AKA a bijection $f:S\rightarrow X$ between $S$ and $X$.

Therefore if we assume that $X$ is infinite then we can suppose that there is a proper $S\subset X$ and a bijection $f:S\rightarrow X$.

Now we can take the proper subset of $Y$ given by the disjoint union $$A=S\cup (Y\backslash X)$$

and map bijectively from $A$ to the disjoint union $X\cup (Y\backslash X)=Y$ and so $Y$ is infinite.

Can you formalise/write down a formula for a bijection $g:A\rightarrow Y$ and can you show that $A$ is indeed a proper subset of $Y$?

score 0 · Answer 2 · answered Oct 02 '13 at 12:32

Depending on the things that you are allowed to assume the proof would differ. If you are allowed to assume that $A \subset B \Rightarrow |A|\leq|B|$, then the proof would be a simple ad absurdum proof - just assume that $Y$ is finite and you reduce the problem to cardinal numbers comparison.

The aforementioned lemma is extremely easy to prove, and the contradiction reveals itself as an implication that an infinite cardinal is lesser or equal to a finite cardinal.

How to prove that if $X$ is a subset of $Y$ , and $X$ is inﬁnite, then $Y$ is inﬁnite

2 Answers2