ArrayFormula and "AND" Formula in Google Sheets

AND doesn't work that way with Array formulae because it ANDs the whole array together in the top left cell, regardless of number of dimensions.

I.e. it checks if "">"" which is FALSE, ANDed with anything it will return FALSE for the top left cell, that result is carried down.

You can use multiplication of truth values to create ANDing that works with ARRAYFORMULA like this:

=ArrayFormula((A1:A>1)*(B1:B>6) = 1)

The OR equivalent would obviously be

=ArrayFormula((A1:A>1)+(B1:B>6) > 0)

=ARRAYFORMULA(IF(ISNUMBER(G3:G),IF(AND( G3:G>=7.5,G3:G<=8),"Full Day",IF(AND(G3:G>8,G3:G<24) ,"Full Day+",IF(AND(G3<7.5,G3>=4),"Half Day",IF(G3<4,"Short Leave",)))),))

Since you use the ARRAYFORMULA function you should use * instead of the AND function.

=ARRAYFORMULA(IF(ISNUMBER(A3:A),IF((A3:A>=7.5)*(A3:A<=8),"Full Day",IF((A3:A>8)*(A3:A<24) ,"Full Day+",IF((A3<7.5)*(A3>=4),"Half Day",IF(A3<4,"Short Leave",)))),))

When using the ARRAYFORMULA function you should use * instead of the AND function and + instead of the OR function.

Explanation

I don't recall an official site about this right now. In any case.

Point 1
Within an arrayformula, AND gives a single value. Not an array of TRUE/FALSE.

Point 2
You must also remember that in "math language", TRUE=1 and FALSE=0

Meaning

+----------+--------+
| Formula  | Result |
+----------+--------+
| =TRUE+2  |      3 |
| =FALSE+2 |      2 |
+----------+--------+

As you can see one can interchange between boolean TRUE/FALSE and 1/0 numbers.

Point 3
About the AND function

The AND function returns true if all the provided arguments are logically true and false if any of the provided arguments are logically false.

Putting it all together
In an arrayformula, instead of using AND/OR when making comparisons, we take advantage of the above information.

So, the "multiplication" (A3:A>=7.5)*(A3:A<=8) will return 1 (meaning TRUE) only if both sides return TRUE. All other conditions return 0 (meaning FALSE).
This is the exact behaviour of the AND function and does work in an ARRAYFORMULA.

About the OR function

The OR function returns true if any of the provided arguments are logically true and false if all of the provided arguments are logically false.

The same logic is applied within the arrayformula, when using + ("addition") instead of the OR function.

AND doesn't go with an Arrayformula. However, there is an interesting way to use AND and OR operators without really using them.

While using arrayformula, in order to do an AND function you must multiply the 2 conditions, for example

=arrayformula(if((condition1)*(condition2), value if true, value if false)) 

Similarly using OR functions is identical EXCEPT instead of a "*" Symbol you would ADD the 2 conditions.

=arrayformula(if((condition1)+(condition2), value if true, value if false)) 

Why is ArrayFormula not repeated for all cells in the column?

As addressed in previous answers,

AND, ANDs the whole array, without regarding dimensions. But, given the recent updates in spreadsheets, using AND respecting dimensions, is now possible. If you want AND, by row, you can use BYROW:

=BYROW(
  A2:B4, 
  LAMBDA(
    row,
    AND(INDEX(row,0,1)>5,INDEX(row,0,2)>6))
)

Here, INDEX is used to access the first column and second column of every row. If you don't want to use INDEX, Another alternative is MAP , which provides a MAP of every argument before LAMBDA:

=MAP(A2:A4, B2:B4, LAMBDA(a,b,AND(a>5,b>6)))