JPARK 's question at Yahoo Answers (Cardinality)

[Fernando Revilla]

Here is the question:

My professor wasn't clear in lecture today, so I'm not exactly sure how I should show this...Can anyone help?

Let A,B,C be sets. Show that if |A| ≤ |B| and |B| ≤ |C|, then |A| ≤ |C|

Here is a link to the question:

Cardinality of Sets Homework Problem? - Yahoo! Answers

I have posted a link there to this topic so the OP can find my response.

 

[Fernando Revilla]

Hello JPARK, $$|A|\leq |B|\Leftrightarrow \exists f:A\to B\mbox{ injective}\\
|B|\leq |C|\Leftrightarrow \exists g:B\to C\mbox{ injective}$$ But $g\circ f:A\to C$ is injective (the composition of injective maps is injective), hence $|A|\leq |C|$.