Plotting XOR In 2D

Definition

Plotting XOR in 2D means treating each input pair as a point.

Input:

[x1, x2]

Coordinates:

x1 = horizontal position
x2 = vertical position

XOR inputs can be plotted as four points in a square.

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The Four Points

XOR has four possible inputs:

[0, 0]
[0, 1]
[1, 0]
[1, 1]

These are the four corners of a square.

Each point has a class label:

[0, 0] -> 0
[0, 1] -> 1
[1, 0] -> 1
[1, 1] -> 0

XOR places labels on the corners of a 2D square.

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Text Plot

One way to view it:

x2
1     class 1: [0,1]     class 0: [1,1]

0     class 0: [0,0]     class 1: [1,0]

      0                  1            x1

The class 1 points are on one diagonal.

The class 0 points are on the other diagonal.

XOR creates a diagonal class pattern.

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What A Linear Classifier Needs

A linear classifier needs one line that separates the classes.

In 2D, the boundary is:

w1*x1 + w2*x2 + b = 0

One side of the line is class 1.

The other side is class 0.

A linear classifier gets one straight line.

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Why The Plot Matters

Looking at the plot reveals the problem.

Class 0 is not all on one side.

Class 1 is not all on one side.

They alternate across the square.

Any straight line that separates one positive point tends to put the other positive point with a negative point.

The 2D plot makes the failure visible.

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Diagonal Separation Is Not Linear Separation

The classes are separated by diagonals as a pattern.

But a single straight line cannot put both class 1 points on one side and both class 0 points on the other.

The issue is not that the points are hard to list.

The issue is the geometry of their arrangement.

XOR is simple logically, but not linearly separable geometrically.

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Why This Matters

The plot connects truth tables to geometry.

The truth table tells the rule.

The plot tells whether a linear boundary can represent it.

For XOR:

truth table is simple
linear boundary is insufficient

Plotting shows the representational limit of one linear classifier.

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Final Summary

• XOR inputs can be plotted in 2D.
• The four inputs are corners of a square.
• Class 1 points are opposite corners.
• Class 0 points are the other opposite corners.
• A linear classifier has one straight boundary.
• No single straight line separates the XOR classes.
• The plot exposes linear inseparability.