`any` and `all` in Python¶

(for Evan)

In [4]:

some_true = [0, 1]
all_true = [1, 1]

all returns True if "all items are True", otherwise False.

In [12]:

all(all_true)

Out[12]:

True

If any item is False, all returns False

In [11]:

all(some_true)

Out[11]:

False

What about empty collections?

In [7]:

all([])

Out[7]:

True

It's ambiguous whether all of an empty collection should be considered true or false. However, this more precise definition should cear things up:

all returns False if any items in the collection are Falsey, True otherwise.

This can be further clarified if we reimplement all with any, which more clearly returns True if a single True value is seen. That's more clearly False if there are no items:

In [17]:

any([])

Out[17]:

False

In [18]:

any(some_true)

Out[18]:

True

If we implement all with any, we first construct the "any items are False" statement, by computing any of the inverse of each item. Then we invert that.

In [9]:

def all_from_any(collection):
    any_false = any(not item for item in collection) 
    return not any_false

Here, any_false is clearly False for an empty collection, and since we return not any_false, all is True for an empty collection:

In [10]:

all_from_any([])

Out[10]:

True

In [20]:

all_from_any(some_true)

Out[20]:

False

In [21]:

all_from_any(all_true)

Out[21]:

True

We can express that in terms of De Morgan's Law, which you might have learned in Boolean Logic:

$$ \overline{A \cdot B} \equiv \overline {A} + \overline {B} $$

which, after inverting both sides, is exactly what our all_from_any above does:

$$ A \cdot B \equiv \overline{ \overline {A} + \overline {B} } $$

Ultimately, it's a choice what to return for empty collections.

any and all in Python¶

`any` and `all` in Python¶