Yes, your alarm will go off (sound) when you set it to go off if your phone is on silent. However, for this to happen you have to make sure that the alarm volume is not on silent.
There are different kinds of volumes on your phone and the alarm volume is usually different from the other volumes. In fact, the alarm volume is given preference over other volumes in your phone.
When you put your phone on silent, it doesn't necessarily affect the alarm volume and for that matter, your alarm will sound even if your phone is on silent.
But if you explicitly turn down the alarm volume or silent the alarm volume on your phone then you may not hear your alarm sound.
Hence, to ensure that your alarm still goes off whiles your phone is on silent, check the alarm volume and make sure it hasn't been silenced.