Q: when using match can I delete the songs from my mobile device to save memory and still listen to all my music from the cloud?

Yes, you can delete them from your device - but if you want to listen to your music on your device then you will need it to be online, so either connected to your carrier if it's a cellular device (which, if you have a limited data allowance, you may not want to do) or be connected to a wifi network.

Using iTunes Match on an iOS device : http://support.apple.com/kb/HT5637