: Which one and why: (alphabets and integers sets) OR (alphabets set and integers set)? Consider the two phrases: alphabets and integers sets alphabets set and integers set Are both grammatically correct? If so, which one

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The use of "alphabets" is a bit odd. You likely mean "letters" [A, B, C...], unless you literally mean sets of alphabets, as in [Latin, Cyrillic...].

Then you're distinguishing between:
letter sets and integer sets [a,b,c] and [1,2,3]
OR
letter and integer sets [a,1,b,2,c,3]

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My first impression is that both are wrong and that what you want to say is "alphabet and integer sets".

Why "Alphabet and Integer Sets"...

A set of alphabets, aka "alphabet sets", is a container of alphabets. It's a grouping, or collection, that holds only alphabets and can contain any number of alphabet sets including zero and approaching infinity.

Logically, it would look something like this:

And the integer set would look like:

However...

If you mean to pluralize the alphabets and integers, then you're probably looking at something like this:

A set of alphabet sets
A set of integer sets

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Speaking as a mathematician, if you mean a set of some, but not all integers, I would say "a subset of the Integers". If however, you mean the set of all the integers, then say "the set of Integers".