The two macros \setminus and \backslash both produce a backslash in mathmode that is customarily used to denote, well, set differences: $A\setminus B$, $A\backslash B$.
For \backslash there are also a \bigger versions, e.g., \big\backslash, whereas \big\setminus creates an error.
Why is that so?
Should we not consider this a bug in \setminus?
The difference is in their definition in fontmath.ltx:
\DeclareMathDelimiter{\backslash} % for double coset G\backslash H
{\mathord}{symbols}{"6E}{largesymbols}{"0F}
\DeclareMathSymbol{\setminus}{\mathbin}{symbols}{"6E}
or, in plain.tex,
\def\backslash{\delimiter"26E30F } % for double coset G\backslash H
\mathchardef\setminus="226E % for set difference A\setminus B
Note that if you do texdef -t latex backslash and texdef -t latex setminus you get
\backslash:
macro:->\delimiter "026E30F
\setminus:
\mathchar"226E
so there's no real difference between the two cases.
There's a substantial definition between a delimiter and a mathchar. The former type will honor \big and friends or \left and \right, the latter won't.
Note that \backslash is defined to be class 0 (ordinary symbol) when used alone, whereas \setminus has class 2 (binary operation symbol); this will affect spacing.
If you need a “bigger \setminus” you can do
\mathbin{\big\backslash}
but \big\setminus would be like \big+ that's not going to work, because \big needs a delimiter after it, that is, either a character with a nonzero \delcode or a macro expanding to \delimiter<27 bit number>.
The usage of \setminus is very different from / that, traditionally, has no space on either side and so TeX treats it as an ordinary symbol, also assigning it a nonzero \delcode
\DeclareMathDelimiter{/}{\mathord}{operators}{"2F}{largesymbols}{"0E}
\DeclareMathSymbol{/}{\mathord}{letters}{"3D}
in LaTeX, and
\mathcode`\/="013D
\delcode`\/="02F30E
in plain TeX.
\big; why should\setminusbehave differently? – egreg Dec 08 '15 at 15:45\big/or\Bigg/? – Steven B. Segletes Dec 08 '15 at 15:47