RETURN
edited 4/14/12

Recognizing Isomorphisms

Which of the following diagrams are isomorphic with which? Why?
 

diagram 1

diagram 2

diagram 3

The letters and numbers in the diagrams (the variables) are commonly called vertices (singular, "vertex"). The connecting lines or arrows (the relations) are commonly called "edges." If this terminology is new to you, or off-putting, think of them as "islands" connected with "bridges."

Diagram 1 (D1) is isormorphic to both D2 and D3 IF

a. we interpret the connecting lines and arrows as merely indicating the relationship "adjacent to" and

b. we map the vertices of D1, a, d, c, and b, respectively, into D2 and D3 to vertices 1, 7, 5, and 3 in both.

Note that if we take the arrows in D1 and D3 to indicate the relationship, "is followed (counterclockwise) by" then D1 and D3 are not isomophic and neither is D1 nor D3 isomorphic to D2.

There are many interpretations possible of both the vertices and edges.

Problem Set A:

1. Can you construct an isomorphism between the English alphabet and the integers from 1 to 26? (Specify matching rules which create this isomorphism.)

A1. This can be done a many ways, frontwards, backwards, shuffled, and so on. Let's just do it with the alphabet in standard order (frontwards). Map A to 1, B to 2, ..., ..., 26 to Z. The relationship by which they are structured into an isomorphism is "is conventionally followed by."

2. Is your watch face isomorphic to the hours in the day?

A2. If your watchface shows twenty-four hours, yes. The digits from 1 to 24 are the vertices which map to the hours of the day. The relation name for the "edges" -- shown by the positions of the digits on your watchface -- is "is followed by."

If your watch has only twelve digits, you can still construct an isomorphism by distinguishing between the first time a digit is reached (or appears, if it is a digital watch, called x AM) and the second time it is reached (called x PM). As before, the relation between digits is "is followed by."

3. In what ways might even a very good map not be isomorphic to the territory it maps?

The map is not the territory it maps. Maps leave things out, they must, otherwise that would be photographic images of what they map. Maps are drawn to focus on items which serve the purposes of the map user. A very good map for finding hotels along a highway is not likely to indicate the locations of every squirrel's nest along the way.

Comparing Programs and Theories

Problem Set B:

1. Consider the teacher who says he is reinforcing student responses by giving certain grades on the report card. If you consider operant conditioning theory, paying careful attention to the variables it deals with and the relationships among them, especially timing, do you think the teacher is actually following the theory? Why or why not?

B1. To use operant theory one must be able to identify:

a. the class of behavioral variables considered to constitute the operant;

b. the class of presentation stimuli considered to constitute the reinforcer;

c. the class of environmental variables consider to constitute the situation;

d. the schedule of reinforcement;

e. the records which constitute the subject's history of reinforcement.

f. the temporal and spacial proximity of a, b, c.

Since report grades are received by the student at some remove from his or her exhibiting the pertinent operant, the answer to whether the teacher is reinforcing the response is likely no, since the theory does not explain how operants followed by other operants can be reinforced at a distance, temporally and spacially, despite a stream of intervening reinforcers having occurred.

2. Is imprisonment necessarily aversive conditioning?

B2. Since some 12% of released prisoners return regularly to prison, it is more likely that they find being out on their own more aversive than being in the prison environment. Action at a distance is a problem. Some enhancement to basic operant theory, for example, hierarchies of cause and effect, will have to be postulated to account for recidivism.

3. If learning by observation alone is possible, what kinds of variables will a theory that purports to explain such learning require?

B3. Probably some kind of long term memory and also an expectation variable along the lines of Albert Bandura's theory. Also needed will be some kind of theory that postulated nervous rerouting without the need for training.

4. Construct a list of questions you would ask a person who claims his or her research (or training) program is guided by a particular theory.

.

TO TOP