Eh, yes and no. AR comes from Armalite and not Assault Rifle. The M4 Carbine is a family of firearms tracing its lineage back to earlier
carbine M16, all based on the original
AR-15 made by
ArmaLite. The AR-15 was first made in both auto and semi auto configs. Then, the M-16 came out (about the same exact thing, really). The M4 Carbine is just a shorter AR-15/M-16.
Honestly, there's really not much difference between the guns. They all look the same and most of the parts can be used between them. Most M16s and M4s are full auto/semi/3rnd burst. Most AR-15s are semi auto.