Understanding Sets and Their Application in GIS

 

Lets dive into the world of sets—a fundamental concept in mathematics that plays an important role in many GIS operations.

Sets are not something from basic maths only but they’re also foundational in spatial data analysis, query-building, overlay operations, and more. So understanding sets isn’t just an academic exercise—it’s a skill that improves our ability to think and work spatially in GIS.

🔢 What Is a Set?

A set is simply a collection of distinct objects, considered as a whole. These objects are called elements or members of the set.

We usually write sets using curly brackets { }.
For example:
A = {2, 4, 6, 8} means set A contains the elements 2, 4, 6, and 8.

Sets are unordered and do not contain duplicate items.

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🧰 Common Set Terminologies

Here are some basic terms you need to know:

• Element: An object in a set (e.g., 4 is an element of A)

• Subset: A set where all elements are also in another set
  (e.g., B = {4, 6} is a subset of A)

• Universal Set (U): The set of all possible elements under consideration

• Empty Set (Ø): A set with no elements

• Cardinality: The number of elements in a set
  (e.g., |A| = 4 for A = {2, 4, 6, 8})

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⚙️ Operations on Sets

Just like arithmetic has operations (+, –, ×), sets also have their own operations:

1. Union ( ∪ )

The union of sets A and B is the set of all elements that are in A, or in B, or in both.
Example:
If A = {1, 2, 3}, B = {3, 4, 5}, then
A ∪ B = {1, 2, 3, 4, 5}

2. Intersection ( ∩ )

The intersection of A and B is the set of elements that are common to both.
Example:
A ∩ B = {3}

3. Difference ( A – B )

This is the set of elements in A but not in B.
Example:
A – B = {1, 2}

4. Complement ( Aʹ )

All elements not in A but in the universal set U.
If U = {1, 2, 3, 4, 5}, and A = {1, 2}, then
Aʹ = {3, 4, 5}

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🗺️ Application of Sets in GIS

Now, let’s connect this to GIS. In GIS, spatial data is often categorized into sets based on location, attributes, or relationships. Set operations help us analyze and manipulate these categories.

Here’s how:

✅ 1. Attribute Queries

Suppose you have a dataset of cities. You want to find cities in the set:

• A = cities with population > 100,000

• B = cities in Gilgit-Baltistan

Then:

• A ∩ B gives cities in GB with population > 100,000

• A ∪ B gives all cities that either have population >100,000 or are in GB

• A – B gives cities with >100,000 population but NOT in GB

These queries are used in GIS tools like QGIS or ArcGIS using SQL or selection tools.

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✅ 2. Spatial Analysis

Think of two vector layers:

• Layer A = flood zones

• Layer B = residential buildings

Using set operations:

• A ∩ B = buildings in flood zones

• B – A = buildings NOT in flood zones

This is exactly how overlay analysis works in GIS. Tools like “Intersect,” “Erase,” “Union,” and “Identity” in GIS software are built on these set concepts.

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✅ 3. Buffer and Proximity

• Let’s say you buffer a road by 500 meters and create a polygon set called A.

• You have a set of hospitals B.

Then:

• A ∩ B = hospitals within 500 meters of the road

• B – A = hospitals farther than 500 meters

This is essential for infrastructure planning, emergency response, and accessibility studies.

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💡 Visualizing Set Operations

In classrooms, we use Venn diagrams to represent sets. In GIS, our layers and maps are the real-world version of Venn diagrams.

When you overlay two spatial layers, you’re doing a visual and computational version of a Venn diagram, combining, separating, or intersecting spatial sets.

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📊 Example Case Study: Agricultural Land Suitability

Suppose you’re identifying suitable farmland using these layers:

• Set A = Areas with slope < 10°

• Set B = Areas with good soil quality

• Set C = Areas within 2 km of a river

The final suitable area = A ∩ B ∩ C
In GIS, you can intersect all these layers to create a map of ideal farmlands.

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🧠 Summary and Key Takeaways

• Sets are basic collections of elements and form the foundation for logical and spatial reasoning.

• GIS uses set theory principles in many operations: selection, overlay, spatial joins, buffer analysis, and query building.

• Understanding union, intersection, difference, and complement helps you work more efficiently in GIS software.

• Tools like Intersect, Union, and Erase in ArcGIS or QGIS are practical applications of set theory.

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🎯 Practice Activity

Try this on your own:

1. Load a land cover layer and a flood hazard map in QGIS.

2. Use "Intersect" to find land cover types affected by flooding.

3. Use "Difference" to find safe land cover areas.

This is you using set operations in real GIS work!

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