### 4. Masses and sets

The following cmavo are discussed in this section:

```     le'i    LE                  the set described as
lo'i    LE                  the set of those which really are
la'i    LA                  the set of those named
```

Having said so much about masses, let us turn to sets. Sets are easier to understand than masses, but are more rarely used. Like a mass, a set is an abstract object formed from a number of individuals; however, the properties of a set are not derived from any of the properties of the individuals that compose it.

Sets have properties like cardinality (how many elements in the set), membership (the relationship between a set and its elements), and set inclusion (the relationship between two sets, one of which — the superset – contains all the elements of the other — the subset). The set descriptors “le'i”, “lo'i” and “la'i” correspond exactly to the mass descriptors “lei”, “loi”, and “lai” except that normally we talk of the whole of a set, not just part of it. Here are some examples contrasting “lo”, “loi”, and “lo'i”:

```4.1)   lo ratcu cu bunre
One-or-more-of-those-which-really-are rats are-brown.
Some rats are brown.

4.2)   loi ratcu cu cmalu
Part-of-the-mass-of-those-which-really-are rats are-small.
Rats are small.

4.3)   lo'i ratcu cu barda
The-set-of rats is-large.
There are a lot of rats.
```
The mass of rats is small because at least one rat is small; the mass of rats is also large; the set of rats, though, is unquestionably large — it has billions of members. The mass of rats is also brown, since some of its components are; but it would be incorrect to call the set of rats brown — brown-ness is not the sort of property that sets possess.

Lojban speakers should generally think twice before employing the set descriptors. However, certain predicates have places that require set sumti to fill them. For example, the place structure of “fadni” is:

```      x1 is ordinary/common/typical/usual in property x2 among the members of set x3
```
Why is it necessary for the x3 place of “fadni” to be a set? Because it makes no sense for an individual to be typical of another individual: an individual is typical of a group. In order to make sure that the bridi containing “fadni” is about an entire group, its x3 place must be filled with a set:
```4.4)   mi fadni zo'e lo'i lobypli
I am-ordinary among the-set-of Lojban-users.
I am a typical Lojban user.
```
Note that the x2 place has been omitted; I am not specifying in exactly which way I am typical — whether in language knowledge, or age, or interests, or something else. If “lo'i” were changed to “lo” in Example 4.4, the meaning would be something like “I am typical of some Lojban user”, which is nonsense.