The final three JOI cmavo, jo'e, ku'a, and pi'u, are probably only useful when talking explicitly about sets. They represent three standard set operators usually called union, intersection, and cross product (also known as Cartesian product). The union of two sets is a set containing all the members that are in either set; the intersection of two sets is a set containing all the members that are in both sets. The cross product of two sets is the set of all possible ordered pairs, where each ordered pair contains a single element from the first set followed by a single element from the second. This may seem very abstract; hopefully, the following examples will help:

Example 14.125.

 lo'i ricfu ku jo'e lo'i dotco cu barda The-set-of rich-things union the-set-of German-things is-large.

Example 14.126.

 lo'i ricfu ku ku'a lo'i dotco cu cmalu The-set-of rich-things intersection the-set-of German-things is-small.

There is a parallelism between logic and set theory that makes Example 14.125 and Example 14.126 equivalent respectively to:

Example 14.127.

 lo'i ricfu ja dotco cu barda The-set-of (rich-things or German-things) is-large.

and

Example 14.128.

 lo'i ricfu je dotco cu cmalu The-set-of (rich-things and German-things) is-small.

The following example uses se remei, which is a set (not a mass) of two elements:

Example 14.129.

 la djeimyz. ce[bo] la djordj. pi'u That-named James and-set that-named George cross-product
 la meris. cebo la martas. cu prami se remei that-named Mary and-set that-named Martha are-lover type-of-pairs.

means that each of the pairs James/Mary, George/Mary, James/Martha, and George/Martha love each other. Therefore it is similar in meaning to Example 14.121; however, that example speaks only of the men loving the women, not vice versa.

Joiks may be combined with bo or with ke in the same way as eks and jeks; this allows grouping of non-logical connections between sumti and tanru units, in complete parallelism with logical connections:

Example 14.130.

 mi joibo do ce la djan. joibo la djein. (I massed-with you) and (that-named John massed-with that-named Jane)
 cu gunma se remei are-a-mass type-of-two-set

asserts that there is a set of two items each of which is a mass.

Non-logical connection is permitted at the joint of a termset; this is useful for associating more than one sumti or tagged sumti with each side of the non-logical connection. The place structure of casnu is:

casnu the mass x1 discusses/talks about x2

so the x1 place must be occupied by a mass (for reasons not explained here); however, different components of the mass may discuss in different languages. To associate each participant with his or her language, we can say:

Example 14.131.

 mi ce'e bau la lojban. pe'e joi (I [plus] in-language that-named Lojban [joint] massed-with
 do ce'e bau la gliban. nu'u casnu you [plus] in-language that-named English ) discuss.

Like all non-logical connectives, the usage shown in Example 14.131 cannot be mechanically converted into a non-logical connective placed at another location in the bridi. The forethought equivalent of Example 14.131 is:

Example 14.132.

nu'i joigi mi bau la lojban gi do bau la gliban. nu'u casnu

Non-logical forethought termsets are also useful when the things to be non-logically connected are sumti preceded with tense or modal (BAI) tags:

Example 14.133.

 la djan. fa'u la frank. cusku That-named John respectively-with that-named Frank express
 nu'i bau la lojban. [start-termset] in-language that-named Lojban
 nu'u fa'u bai [joint] respectively-with under-compulsion-by
 tu'a la djordj. [nu'u] something-about that-named George.
 John and Frank speak in Lojban and under George's compulsion, respectively.

Example 14.133 associates speaking in Lojban with John, and speaking under George's compulsion with Frank. We do not know what language Frank uses, or whether John speaks under anyone's compulsion.

Joiks may be prefixed with i to produce ijoiks, which serve to non-logically connect sentences. The ijoik .ice'o indicates that the event of the second bridi follows that of the first bridi in some way other than a time relationship (which is handled with a tense):

Example 14.134.

 mi ba gasnu la'e di'e .i I [future] do the-referent-of the-following:
 tu'e kanji lo ni cteki .ice'o lumci le karce ( Compute the quantity-of taxes. And-then wash the car.
 .ice'o dzukansa le gerku tu'u And-then walkingly-accompany the dog. )
 List of things to do: Figure taxes. Wash car. Walk dog.

Example 14.134 represents a list of things to be done in priority order. The order is important, hence the need for a sequence connective, but does not necessarily represent a time order (the dog may end up getting walked first). Note the use of tu'e and tu'u as general brackets around the whole list. This is related to, but distinct from, their use in Section 14.8, because there is no logical connective between the introductory phrase mi ba gasnu la'edi'e and the rest. The brackets effectively show how large an utterance the word di'e, which means the following utterance, refers to.

Similarly, .ijoi is used to connect sentences that represent the components of a joint event such as a joint cause: the Lojban equivalent of Fran hit her head and fell out of the boat, so that she drowned would join the events Fran hit her head and Fran fell out of the boat with .ijoi.

The following nai, if present, does not negate either of the things to be connected, but instead specifies that some other connection (logical or non-logical) is applicable: it is a scalar negation:

Example 14.135.

 mi jo'u nai do cu remei I in-common-with [not!] you are-a-twosome

The result of mi jo'u do would be two individuals, not a mass, therefore jo'u is not applicable; joi would be the correct connective.

There is no joik question cmavo as such; however, joiks and ijoiks may be uttered in isolation in response to a logical connective question, as in the following exchange:

Example 14.136.

 do djica tu'a loi ckafi You desire something-about a-mass-of coffee
 ji loi tcati [what-connective?] a-mass-of tea?
 Do you want coffee or tea?

Example 14.137.

 joi Mixed-mass-and.
 Both as a mass (i.e, mixed together).

Ugh. (Or in Lojban: .a'unaisairo'o.)