If she is my friend and you are her friend, then we are friends. Given this,…

A binary relation R on a set can have several named properties. R is transitive when a R b and b R c together imply a R c. R is symmetric when a R b implies b R a. R is reflexive when a R a holds for every element a. R is an equivalence relation only when it is simultaneously reflexive, symmetric, and transitive. An implication is simply the logical if-then form of a statement, and is not by itself a named property of a relation.

The stem's rule states: if she is my friend (a R b) and you are her friend (b R c), then we are friends (a R c). This has exactly the form a R b and b R c implies a R c, which is the transitive property applied to the friend relation. Nothing in the stem shows that the friend relation also holds in the reverse direction (symmetric), or that a person is trivially their own friend (reflexive). So the property the stem actually demonstrates is transitivity, and transitivity alone.

• (i) and (ii) pairs the transitive chaining pattern named in item (ii) with the commutative label from item (i). Commutativity is a term normally applied to operations such as addition (a + b = b + a), and would require the friend relation to also be shown holding in a swapped order -- a separate claim from the chaining described in the stem.

• (iii) selects only the implicative label from item (iii). An implicative structure describes the general if-then shape shared by any conditional sentence -- a feature of the sentence's phrasing rather than a claim about how the three people (self, her, you) are specifically chained together.

• (i), (ii), (iii) and (iv) bundles together all four labels from items (i) through (iv), including the equivalence label from item (iv). An equivalence relation additionally requires that every element relate to itself (reflexivity) and that the relation hold in both directions between any two related elements (symmetry) -- separate claims from the chaining described in the stem.

• None of these is the residual choice that applies whenever the property demonstrated by the stem's rule does not match any of the specific labeled combinations listed among the other three choices.

Because the stem demonstrates transitivity alone, and none of the three combination choices states transitivity alone, the correct choice among the options offered is None of these.