# Behavior of all() and any() on empty vectors

**URL:** <https://discourse.julialang.org/t/behavior-of-all-and-any-on-empty-vectors/30963>\
**Category:** General Usage\
**Created:** [November 11, 2019, 6:12pm UTC](https://discourse.julialang.org/t/behavior-of-all-and-any-on-empty-vectors/30963 "2019-11-11T18:12:57Z")\
**Posts on this page:** 8\
**Page:** 1

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**Author:** ![kevbonham](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/kevbonham/32/216165_2.png) [@kevbonham](https://discourse.julialang.org/u/kevbonham)\
**Post date:** [November 11, 2019, 6:12pm UTC](https://discourse.julialang.org/t/behavior-of-all-and-any-on-empty-vectors/30963/1 "2019-11-11T18:12:57Z")

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This caught me by surprise:

```julia
julia> any(x->true, [])
false

julia> all(x->true, [])
true

julia> any(x->false, [])
false

julia> all(x->false, [])
true

```

This seems like a bug - I think that `all` should never be true when `any` is false. But I’m guessing there’s a principled reason for this. Can anyone explain it to me?

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**Author:** ![mbauman](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/mbauman/32/31082_2.png) [@mbauman](https://discourse.julialang.org/u/mbauman)\
**Post date:** [November 11, 2019, 6:32pm UTC](https://discourse.julialang.org/t/behavior-of-all-and-any-on-empty-vectors/30963/2 "2019-11-11T18:32:58Z")

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The key is this that `any` returns `true` if there are any `true` elements. No elements means that there can’t be any `true` elements, thus it’s always `false`. Similarly `all` returns `false` if there are any `false` elements. No elements means there can’t be any `false` elements, thus it’s always `true`. Both short-circuit upon finding the first “exceptional” value when it knows the result.

Practically, I really like the `any` behavior but — while I see the motivation and symmetry — I often want `all([])` to be `false`. I often just have to remember to add in the fact that I want `!isempty` to my predicates that include `all`.

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**Author:** ![kevbonham](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/kevbonham/32/216165_2.png) [@kevbonham](https://discourse.julialang.org/u/kevbonham)\
**Post date:** [November 11, 2019, 6:42pm UTC](https://discourse.julialang.org/t/behavior-of-all-and-any-on-empty-vectors/30963/3 "2019-11-11T18:42:47Z")

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> [@mbauman](#):
>
> while I see the motivation and symmetry — I often want `all([])` to be `false`

Yeah, this seems more sensible to me. Logically, it seems like `all(f, x) && !any(f, x)` should never be true.

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**Author:** ![DNF](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/dnf/32/10191_2.png) [@DNF](https://discourse.julialang.org/u/DNF)\
**Post date:** [November 11, 2019, 6:42pm UTC](https://discourse.julialang.org/t/behavior-of-all-and-any-on-empty-vectors/30963/4 "2019-11-11T18:42:57Z")

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This behavior agrees with Matlab, at least.

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**Author:** ![mbauman](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/mbauman/32/31082_2.png) [@mbauman](https://discourse.julialang.org/u/mbauman)\
**Post date:** [November 11, 2019, 7:23pm UTC](https://discourse.julialang.org/t/behavior-of-all-and-any-on-empty-vectors/30963/5 "2019-11-11T19:23:39Z")

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Yeah, there are two ways to think about these operations. There’s the way we (and Matlab and Numpy) operate:

- `any`: `true` if contains a `true`, `false` otherwise
- `all`: `false` if contains a `false`, `true` otherwise

They could also potentially be defined by leading with the converse in an alternate universe′:

- `any′`: `false` if _only contains_ `false`s, `true` otherwise. Thus: `any′([]) == true`
- `all′`: `true` if _only contains_ `true`s, `false` otherwise. Thus: `all′([]) == false`

I think colloquially, we tend to think about how things are true (that is, leading with the `true` if…; the `any` and `all′` definitions), but if the “all” operation is going to short-circuit like any does, then it’s sensible to be defined as `all`. The docstring could probably be clearer in this regard.

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**Author:** ![rfourquet](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/rfourquet/32/3610_2.png) [@rfourquet](https://discourse.julialang.org/u/rfourquet)\
**Post date:** [November 11, 2019, 7:54pm UTC](https://discourse.julialang.org/t/behavior-of-all-and-any-on-empty-vectors/30963/6 "2019-11-11T19:54:17Z")

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> [@mbauman](#):
>
> `any′` : `false` if _only contains_ `false` s, `true` otherwise. Thus: `any′([]) == true`

With this definition, it’s not obvious to me that `any'([]) == true`… In a sense, `Bool[]` contains only `false`, as it doesn’t contain any `true`s and can’t contain something else than `true` or `false`.

I don’t know if the current definition is “principled”, but it’s consistent with `sum` and `prod`: `any(a)` is equivalent to `sum(a) % Bool` (think a sum in the 2-element field `GF(2)`), and `all(a)` is equivalent to `prod(a)`. An empty sum is usually defined to be the addition identity, i.e. 0 or `false` in our case, whereas an empty product is defined to be the mutliplication identity, i.e. 1 or `true`.

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<div class="post-metadata">

**Author:** ![mbauman](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/mbauman/32/31082_2.png) [@mbauman](https://discourse.julialang.org/u/mbauman)\
**Post date:** [November 11, 2019, 8:02pm UTC](https://discourse.julialang.org/t/behavior-of-all-and-any-on-empty-vectors/30963/7 "2019-11-11T20:02:33Z")

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> [@rfourquet](#):
>
> In a sense, `Bool[]` contains only `false` , as it doesn’t contain any `true` s and can’t contain something else than `true` or `false` .

I would say `Bool[]` does not contain a single `false`. Perhaps there’s better verbiage to be used than “only contains”, but it’s tricky to define these without using the words (implicitly or explicitly) themselves. I think you’re implicitly using the “`true` if” definition here. 🙂

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<div class="post-metadata">

**Author:** ![rfourquet](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/rfourquet/32/3610_2.png) [@rfourquet](https://discourse.julialang.org/u/rfourquet)\
**Post date:** [November 11, 2019, 8:17pm UTC](https://discourse.julialang.org/t/behavior-of-all-and-any-on-empty-vectors/30963/8 "2019-11-11T20:17:14Z")

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> [@mbauman](#):
>
> I think you’re implicitly using the " `true` if" definition here

Absolutely! It’s because I can’t really decide if an empty collection contains “only false”… it’s not a clear concept, so for me the closest definition to disambiguate this corner case is the “true if”, while for you it’s the “not empty and contains only false” 🙂