#### 3.2 Dicing

Dicing (it may be called also down-sizing) a n-dimensional datacube means to select a (possibly
empty) subset of members from each dimension and discard all tuples which contains at least
one member not in the selection (see Figure 3

Figure 3: Dicing a Cube

Note that the resulting datacube has the same dimension as the datacube before applying
the operation. Even if the selection on a particular dimension contains a single member, it is not
the same as slicing. The dimension is not reduced and a set containing a single member after all
is also a set.

If no member is selected of any dimension, we get the empty cube. (Note that the empty
cube is not the same than a 0-dimensional cube, which contains a single cell.) If all
members are selected from all dimensions, the resulting cube is the same as the original
cube.

Mathematically, the operation is simply stated as follows. As above, let’s start again with
the 3-dimensional cube a_{i,j,k}. Let P_{i,j,k} be a predicate (relation) which is true for some
(i,j,k)-triples and false for the others. Then dicing a_{i,j,k} on P_{i,j,k} means to create a cube d_{i,j,k}
that contains all the (i,j,k)-triples of a_{i,j,k} for which P_{i,j,k} is true. The result is an
assignment:

A shorter notation is :