Set theory, Systems and Jaga Mission

In his classic book 'A System's View of Planning', George Chadwick wrote:

"Not only can the whole of mathematics be developed from the concept of a set, but, as foreshadowed, the idea of a system stems naturally from that of a set." (p-28)

While we all studied set theory in high school mathematics, its usefulness in making sense of the structure and behaviour of complex systems encountered regularly in urban planning, was never discussed adequately in planning school. The consequence is the absence of yet another powerful tool from the contemporary planners' toolkit and the state of confoundedness that naturally follows.

Created by the German mathematician Georg Cantor in 1874, set theory "stems from the simple idea of a number of things which have a common property or properties and thus can be represented as elements of a set." (ibid)

The relationship of set theory with the systems view of planning is made amply clear when we consider that, "the commonly accepted definition of a system is a set of entities and the relationships between them."

Regions and Sets

Let us consider how set theory helps us to tackle the complexity in Jaga Mission, the flagship slum land-titling and upgrading project of the Government of Odisha, India. But before plunging into that, let's have a quick look at how set theory came to be an integral part of regional science already by the 1960s. 

In his classic paper, 'Mathematical Aspects of the Formalization of Regional Geographic Characteristics' , the Soviet geographer B.B. Rodoman wrote that, if a region is viewed as a set of subregions, then one could "convert into the language of geography the theorem of the five alternative relationships which is part of set theory." 

He elaborated further that, according to set theory, two regions A and B may have the following relationships with each other:

1) They may have no common territory

2) They may intersect

3) A may be part of B

4) B may be part of A

5) They may be identical

The relationships can be expressed as follows by using the symbols of set theory:

1) A ∩ B = ∅    [intersection of A and B is a null set]

2) A ∩ B ≠ ∅ ; A ∩ B ≠ A ; A ∩ B ≠ B   [intersection of A and B is not a null set]

3) A ∩ B ≠ ∅ ; A ∩ B = A ; A ∩ B ≠ B ; A ⊂ B   [A is a sub-set of B]

4) A ∩ B ≠ ∅ ; A ∩ B ≠ A ; A ∩ B = B ; B ⊂ A    [B is a sub-set of A]   

5) A ∩ B ≠ ∅ ; A ∩ B = A ; A ∩ B = B ; A = B    [A is equal to B]

By adding to the above the relationships of the sets with their complements (i.e. the elements present in the universal set but not in the set itself - basically the world outside of itself), one can show the full range of ways in which various overlapping or separated regions interact with each other. This was explained very clearly through an example of wheat growing regions, vegetable growing regions and corn growing regions in Golledge and Amadeo's paper titled 'Some introductory notes on regional division and set theory'

It is clear from the diagram above that every part of the three fields, no matter how complex, could be accurately described using the language of sets. For example parts 4 and 6, which occupy the central part of the fields, where all three type of fields intersect can be described using the following notations -

For part 4 --> (W ∩ V) ∪ (W ∩ C)        

[i.e. the union of the intersection of wheat and vegetable and the intersection of wheat and corn]

For part 5 --> (C ∩ W) ∪ (C ∩ V)

[i.e. the union of the intersection of corn and wheat and the intersection of corn and vegetable]

It is easy to spot the origins of the various vector operations in GIS using logical operations such as AND, OR, != (corresponding to intersection, union and not equal to) etc from the above discussion on set theory and regionalization.  

Slums and Sets 

Any slum land titling project is complex by its very nature, but Jaga Mission is quite the Godzilla of complexity due to its size and geographical coverage. Unlike, slum titling and upgrading projects that target a couple of major cities, the Mission covers all 2919 slums in all 115 cities and towns in the state.

However, by combining the necessary geo-spatial datasets corresponding to the various operational parameters of the mission one can readily apply set theory to simplify and automate the tasks. This was particularly true in the case of the trickiest component of any land titling project - the land parcels themselves. 

In fact, one is bound to spot the visual similarity in the following image of a slum of Jaga Mission shown below and the illustrative diagram of the three fields in Golledge and Amadeo's paper.

The above map shows the location of slum houses overlaid on land parcels which belong to three types - Leasable government land (on which slum land rights can be granted); Reserved government land (on which slum land rights can be granted only after a category conversion process); and Private land (on which slum land rights cannot be granted).

If A is the set of slum houses and B is the set of government leasable land parcels then the slum houses entitled to land titles straight away would be given by - 

A ∩ B    

However, if one would consider the total set of slum houses which are entitled to land titles once the land category conversion for reserved government land parcels are completed (reserved parcels given by set C), then that would be given by -

A ∩ (B ∪ C)

If private land parcels are the only category over which land titles cannot be granted (set D) then the set of entitled slum houses could also be given by -

A ∩ D'    [where D' is the complement of set D]

By defining the sets according to the specific parameters of the mission, the outcome of the interaction of various parameters could be computed by applying the theorem of alternate relationships.

Once such relationships are established then it really does not matter if the process needs to be done for one slum or for a 100 slum or for a 1000 slums. Nor is it any difficulty to divide a particular set into its constituent sub-sets (for example the reserved government land category itself is a union of numerous subsets of land parcel types distinguished by the land-use type and the ownership type -- these particulars can also be described as sets of their own).