Conventions regarding the use of space in mathematical expressions [duplicate]

This question already has an answer here:

The Chicago Manual of Style (sec. 12.16) suggests medium space (a ¼ em) before and after

• signs for binary operations (i.e., conjunctions);
• symbols of integration, summation, or union; and
• signs for binary relations (i.e., verbs)

except in exponents; also medium space after commas between "coordinate points" or between elements in a list; and "no space after a binary operation or relation sign when it is modifying a symbol (i.e., used as an adjective)".

I don't know whether LaTeX follows these guidelines, but anyway I would like to know the (generally accepted) "right" amount of space in the following situations (that is, do I need to add/remove space, and if so how much?)

• after logarithm: \log x
• before factorial symbol: y!
• in an implicit multiplication: x y
• before and after "for all" symbol: x > 0 \forall x
• between equations separated by a comma: x = 1, y = 2
• between equations separated by text: x = 1 \text{ and } y = 2
• after a comma separating elements of a set: \{0, 1\}
• after a comma separating arguments of a function: f(x, y)

marked as duplicate by Joseph Wright♦Dec 5 '16 at 17:53

• All this is managed by (La)TeX and rarely has to be modified. For quantifiers, note you should write them after the relation, but before. – Bernard Dec 5 '16 at 1:21
• I always put a \quad before and after \text{} in \displaystyle. Also before \forall. – John Kormylo Dec 5 '16 at 1:30
• This, in my opinion, may solicit "primarily opinion-based answers". – Werner Dec 5 '16 at 4:29
• I've closed here as a dupe (rather than opinion based) as we do have question on this topic, as pointed out by @ErnestA! – Joseph Wright Dec 5 '16 at 17:54

after logarithm: $\log x$ -- exactly this space
between equations separated by a comma: $x = 1$,  $y = 2$ -- two different equations, separated by a space
after a comma separating elements of a set: \{0, 1\} -- it depends, some people want to have spaces after commas (e.g. ,\ ).