how can we obtain oxygen and hydrogen at the same time?

圖片參考：http://upload.wikimedia.org/wikipedia/en/4/4c/Hoffman_voltameter.jpg

Electrolysis of Water is an electrolytic process which decomposes water into oxygen and hydrogen gas with the aid of an electric current, where a power source from a 6 volt battery is commonly used. The electrolysis cell consists of two electrodes (usually an inert metal such as platinum) submerged in an electrolyte and connected to opposite poles of a source of direct current.
The electric current disassociates water molecule into hydroxide
圖片參考：http://upload.wikimedia.org/math/7/2/4/724c03d1ced617e31eebe810371f18a8.png
ions.
In the electrolytic cell, at the cathode, hydrogen ions accept electrons in a reduction reaction that forms hydrogen gas:


圖片參考：http://upload.wikimedia.org/math/9/4/f/94fb9bd83f4deeee36c15e49b67b5661.png

At the anode, hydroxide ions undergo an oxidation reaction and give up electrons to the anode to complete the circuit and form water and oxygen gas:


圖片參考：http://upload.wikimedia.org/math/b/b/0/bb01a49f12a150ab1be8cf74db2df23b.png

hence decomposing water into Oxygen and Hydrogen;


圖片參考：http://upload.wikimedia.org/math/a/c/7/ac7f14b66e4b5bd7cd974ccc39140404.png

The volume of hydrogen gas produced is therefore twice the amount of oxygen gas. Assuming equal temperature and pressure for both gases, the hydrogen gas has twice the volume of the oxygen.
Extracted from the following web-site:
http://en.wikipedia.org/wiki/Electrolysis_of_water

I hope I can help you.