Yes having attributes containing the union of the sets of attributes {“can read”} and {“can write”} means being in the intersection of the set of things having the attribute “can read” and that having the attribute “can write”. Set union, intersection, and membership are related that way.

(Which means saying that “and” is about intersection and not union is nonsense, everything statement about set union can be rephrased to an equivalent statement about intersection, and vice versa.)

Yeah, there's some duality there. If each predicate is identified with the set of things that satisfy that predicate, then the "and" of two predicates is identified with the intersection of the sets that satisfy the two predicated. On the other hand, if you instead use predicates by talking about the set of predicates that objects satisfy, then saying that something satisfies one set of predicates, and satisfies some other set of predicates, then that says that the thing satisfies all the predicates in the union of those two sets.

For some relation R between elements of the set X and elements of the set Y (we can think of X as being the set of individuals, and Y as being a set of elementary unary predicates), for each x in X, we can refer to A(x) := {y in Y : x R y} , and for each y in Y we can talk about B(y) := {x in X : x R y} . When we say (x_1 R y_1) && (x_1 R y_2), this is equivalent to both (x_1 in B(y_1)) && (x_1 in B(y_2)) i.e. (x_1 in (B(Y_1) intersect(B(y_2)))) , and also to (y_1 in A(x_1)) && (y_2 in A(x_1)) , i.e. ({y_1} \subset A(x_1)) && ({y_2} \subset A(x_1)) i.e. (({y_1} union {y_2}) \subset A(x_1)) .

Personally, if we are talking about a single x_1, and we are saying "and" about a number of different y_1 , y_2 , etc. , or if we are talking about each x in some collection but are saying the same thing about each of them, namely that for each x in the collection, that x R y_1 and that x R y_2 and that x R y_3 , I think it seems more natural to take the first of the two framings described above. That is, I think it seems more natural to say that for each x in the collection, that x \in (B(y_1) intersect B(y_2) intersect B(y_3)), rather than saying that ({y_1} union {y_2} union {y_3}) subset A(x) .

Or, if we give the name W to the collection we are talking about, we could say that "W subset (B(y_1) intersect B(y_2) intersect B(y_3))". This seems to me a fairly natural way to say "all the W , y_1 , y_2, and y_3" ("each of my 3 pokemon can do reading, writing, and arithmetic") .

The other way would end up more like, "({y_1} union {y_2} union {y_3}) subset (the intersection over all x in W, of A(x))".

This works, but seems less natural to me.

Also, especially in the case where you expect X to be substantially larger than Y.