The Fundamental Operation: Projection

To illustrate a further important point, below is an actual object-property table. It shows 10 objects and 5 attributes. You can think of it as representing 10 objects that just came off an assembly line. Each of those objects has 5 properties, for example `\{P_1,P_2,P_3,P_4,P_5\}=\{text{color},text{shape},text{texture},text{size},text{orientation}\}`. Thus, for example, these objects can have three possible colors, `P_1 in \{0,1,2\}`, which might be `\{text{blue},text{red},text{green}\}`. It can be noted that not all of the objects need be distinct, and in fact there are several identical objects in this set. (Can you find them? Just kidding.)

Object	Property
	`P_1`	`P_2	`P_3`	`P_4`	`P_5`
`O_1`	1	2	0	1	1
`O_2`	1	2	0	1	1
`O_3`	2	0	0	1	0
`O_4`	0	0	1	2	1
`O_5`	2	1	0	2	1
`O_6`	0	0	1	2	2
`O_7`	2	0	0	1	0
`O_8`	0	1	2	2	1
`O_9`	2	1	0	2	2
`O_{10}`	2	0	0	1	0

The crucial fact that must be observed is that the distinguishability of objects depends on which attributes are considered. For example, in the table above, we can already note that objects `\{O_1,O_2\}` are indistinguishable, as are `\{O_3,O_7,O_{10}\}`. The objects in these sets correspond to the same point in feature space, even if they are physically distinct objects (which we assume they are). However, if we were to further assume that attributes `\{P_1,P_2,P_3\}` are inaccessible, and were therefore able to consider only attributes `P_4` and `P_5`, we would observe (table below) that several other objects now also become indistinguishable, collapsing to the same point in feature space.

Object	Property
	`P_4`	`P_5`
`O_1`	1	1
`O_2`	1	1
`O_3`	1	0
`O_4`	2	1
`O_5`	2	1
`O_6`	2	2
`O_7`	1	0
`O_8`	2	1
`O_9`	2	2
`O_{10}`	1	0

In particular, we now find that we have the following sets of indistinguishable objects:

`{(\{O_1,O_2\}),(\{O_3,O_7,O_{10}\}),(\{O_4,O_5,O_8\}),(\{O_6,O_9\}):}`

So there are only four discriminable kinds of objects represented in this knowledge system after attributes `\{P_1,P_2,P_3\}` have been dropped and consideration has been restricted to attributes `\{P_4,P_5\}`. These four kinds are called "equivalence classes" because the entities contained in each class are indistinguishable (equivalent) based on the attributes under consideration. The process of "dropping attributes" is called projection and corresponds to an orthogonal (parallel to axes) geometric projection in feature space. In the previous case, we would say that the data is projected onto dimensions `\{P_4,P_5\}`. In general, the projection onto a given set of attributes will produce a collection of equivalence classes containing objects which cannot be disambiguated based on those attributes. This will be extremely important in what follows.