# What do you call the two components resulting from filtering?

Suppose I am given the numbers from 1 to 9:

1 2 3 4 5 6 7 8 9

I filter out the prime numbers:

2 3 5 7 (A)

and I am left with:

1 4 6 8 9 (B)

What do you call (A) and (B) with respect to the filtering?

I am not sure if "filtered out" refers to the component which was removed, or the component which was retained.

Mathematicians use the term residue for anything left over, which fits nicely with the concept of filtering. I would call (B) the residue and (A) the filtrate in your example. The residue in this case, by the way, is the union of the positive composites and the unit 1.

Wikipedia

Oxford Dictionaries

Often also called the remainder, which is listed as a synonym for residue at Oxford dictionaries, but the more usual expression in Mathematics is to call it a residue.

Also, from Mathworld

• residue is used for anything left over, the examples given at Mathworld are just examples.
– JMP
Commented Apr 1, 2015 at 9:06
• I'd remembered the word residue from chemistry class, but didn't remember the word from set theory. Thanks for the refresher!
– J.R.
Commented Apr 1, 2015 at 9:53

"Filtered out" would refer A as it was the set that was removed.

(B) I would call the set "that remains" or something similar. There may be a better word for (B)

• For (B) you could use "remaining numbers" Commented Apr 1, 2015 at 6:31