operating system vs DOS

operating system and DOS both are nouns.

Word	Noun	Adjective	Verb	Adverb
operating system	Yes	No	No	No
DOS	Yes	No	No	No

As nouns, DOS is a hyponym of operating system; that is, DOS is a word with a more specific, narrower meaning than operating system:

operating system: (computer science) software that controls the execution of computer programs and may provide various services
DOS: an operating system that is on a disk

Other hyponyms of operating system include disk operating system, UNIX, UNIX operating system, UNIX system, Windows, android.

operating system (noun)	DOS (noun)
(computer science) software that controls the execution of computer programs and may provide various services	an operating system that is on a disk