Which adjective describes the "either ... or ..." relationship? [duplicate]

Question

Is there an adjective that describes the "either ... or ..." relationship between two concepts?

Here by "either ... or ...", I mean mutual exclusiveness and exhaustiveness. Or logically speaking, the "either ... or ..." relationship can be thought of as the relationship between a proposition and its negation.

Let me use two examples to clarify what I need.

1. Suppose I'm an elementary school student and I just learned about odd and even numbers. I'm don't know if a integer can be both odd and even (multual exclusive), and I don't know if every integer must be odd or even (exhaustiveness). So I go to my teacher and ask "is a integer either odd or even?". Can I rephrase my question like "are odd and even numbers ____?"?
2. Suppose instead I'm a college student now and I just learned about open and closed sets. Misled by the their names, I thought that a set must be open if it's not closed and vice versa, until my professor told me that "open sets and closed sets are not ____".

I want an adjective that fits into those blanks above.

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After reading all the answers and comments, as well as related questions, I think the "either ... or ..." phrase might be the simplest and most suitable way, as least for everyday usage. Just say "an X is either A or B" instead of "A and B are ____"

Answer 1

In mathematical literature, "complete" is the usual term, e.g.

...completeness implies that there are not any “gaps” (in Dedekind's terminology) or “missing points” in the real number line... https://en.wikipedia.org/wiki/Completeness_of_the_real_numbers

The Completeness of the Integers, John McCuan, February 25, 2020 https://people.math.gatech.edu/~mccuan/courses/4317/integers2.pdf

Picking up on the comment by @271828, I would say that "exhaustive" is equally correct in general speech. It doesn't even depend on there being merely two alternatives.

E.g. The digits 0-9 are exhaustive when it comes to expressing the numbers from zero to one hundred.

In other words they exhaust the possibilities. In less formal terms, you could say "all encompassing", however your example is mathematical and so formal by definition.

EDIT

Incidentally, the terms necessary and sufficient are relevant here. For example Even and Odd numbers are sufficient to encompass the integers but they aren't necessary, i.e. you can describe integers in a different way.

Answer 2

Are you looking for the word dichotomous?

Lexico defines dichotomous as

Exhibiting or characterized by dichotomy,

and dichotomy as

A division or contrast between two things that are or are represented as being opposed or entirely different.