Understanding Continuous Mathematics V/S Discrete Mathematics

Continuous Vs Discrete

Understanding the clear difference between continuous and discrete mathematics in depth is important as the first step toward building an LLM from scratch. Mostly, we deal with continuous mathematics approaches, like Linear Algebra, while building models.

Continuous Mathematics and Discrete Mathematics are two fundamental branches of mathematics, each dealing with different types of problems and methods.

Continuous Mathematics: Deals with continuous variables and smooth changes. It includes fields such as calculus, differential equations, and real analysis. Problems often involve quantities that can change fluidly.

Discrete Mathematics: Deals with discrete variables and distinct changes. It includes fields such as combinatorics, graph theory, and logic. Problems often involve countable, separate values.

To highlight the differences, let’s compare how each branch approaches a specific type of problem: finding the shortest path.

Problem: Finding the Shortest Path

Discrete Mathematics Approach

Scenario: Find the shortest path in a graph.

Problem Statement: Given a graph with nodes and edges, where each edge has a weight (representing distance), find the shortest path from a starting node to a target node.

Example: Consider a graph with the following nodes and weighted edges:

   A
  / \
 1   3
/     \
B---2---C
 \     /
  1   1
   \ /
    D

• Nodes: A, B, C, D
• Edges: (A-B, 1), (A-C, 3), (B-C, 2), (B-D, 1), (C-D, 1)

Solution Using Dijkstra’s Algorithm:

1. Initialization:

• Start at node A with distance 0.
• Set distances to other nodes to infinity.

1. Iteration:

• Visit the nearest unvisited node (initially A).
• Update distances to adjacent nodes.
• Repeat until all nodes are visited.

Steps:

• Start at A: Distances are A(0), B(1), C(3), D(∞)
• Visit B: Distances are A(0), B(1), C(3), D(2) (through B)
• Visit D: Distances are A(0), B(1), C(3), D(2)
• Visit C: Distances are A(0), B(1), C(3), D(2)

Shortest Path from A to D: A → B → D with distance 2.

Continuous Mathematics Approach

Scenario: Find the shortest path along a smooth curve.

Problem Statement: Given a continuous function representing a path, find the shortest distance between two points on the curve.

Example: Consider the curve described by the function 𝑦=𝑓(𝑥)y=f(x) between points 𝐴(𝑥1,𝑦1)A(x1​,y1​) and 𝐵(𝑥2,𝑦2)B(x2​,y2​).

Solution Using Calculus:

1. Arc Length Formula: The length of the curve between two points is given by the integral:
2. 𝐿=∫𝑥1𝑥21+(𝑑𝑦𝑑𝑥)2 𝑑𝑥L=∫x1​x2​​1+(dxdy​)2​dx
3. Compute the Derivative:

• Find 𝑑𝑦𝑑𝑥dxdy​ from the function 𝑦=𝑓(𝑥)y=f(x).

1. Integrate:

• Integrate the expression to find the arc length.

Example Calculation: For 𝑦=𝑥2y=x2 between 𝑥=0x=0 and 𝑥=1x=1:

1. Compute 𝑑𝑦𝑑𝑥=2𝑥dxdy​=2x.
2. Integrate:
3. 𝐿=∫011+(2𝑥)2 𝑑𝑥=∫011+4𝑥2 𝑑𝑥L=∫01​1+(2x)2​dx=∫01​1+4x2​dx
4. This integral can be solved using substitution or numerical methods.

Shortest Path: The shortest path is the arc length along the curve between the specified points.

Key Differences

• Nature of the Problem:
• Discrete Mathematics: Deals with distinct, countable elements like nodes and edges in a graph.
• Continuous Mathematics: Deals with continuous elements and smooth changes like curves and surfaces.
• Techniques Used:
• Discrete Mathematics: Uses algorithms and combinatorial methods (e.g., Dijkstra’s algorithm).
• Continuous Mathematics: Uses calculus and analysis (e.g., integration).
• Types of Solutions:
• Discrete Mathematics: Solutions involve paths, sequences, and arrangements.
• Continuous Mathematics: Solutions involve integrals, derivatives, and functions.

Summary

• Discrete Mathematics is ideal for problems involving distinct, separate values like graph traversal, scheduling, and counting.
• Continuous Mathematics is ideal for problems involving continuous change like motion, growth, and area under curves.

Understanding both branches provides a comprehensive toolkit for tackling a wide variety of mathematical and real-world problems.