Boolean operation::not-equal: 'df' vs. 'ne' vs. 'neq'

[rakhimov]

ne seems to be more common as an operator to signify 'not-equal'. Bash comes into mind.
neq is, for example, used in LaTeX.
Where does df come from?

[rakhimov]

One more confusion with df would be its association with difference/without of sets.
That is, df(a, b) = and(a, not(b)) or a\b = a * b'