XOR Classes

Definition

XOR creates two classes.

Class 0:

inputs are the same

Class 1:

inputs are different

XOR classes separate matching inputs from different inputs.

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Class 0

Class 0 contains inputs where both values match.

Examples:

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

Both inputs false:

0, 0

Both inputs true:

1, 1

In both cases, XOR returns 0.

Class 0 means no exclusive difference.

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Class 1

Class 1 contains inputs where the values differ.

Examples:

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

Exactly one input is true.

That is the exclusive OR case.

Class 1 means exactly one input is true.

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Class Pattern

The four points are:

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

So:

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

The classes are crossed.

They do not sit cleanly on one side of a line.

XOR classes alternate across the square.

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Why This Is Different From Simple Separation

Some classification problems put all class 0 examples on one side and all class 1 examples on the other.

XOR does not.

Class 0 appears in two opposite corners.

Class 1 appears in the other two opposite corners.

This is why one straight boundary cannot separate them.

XOR classes are diagonally separated, not side-separated.

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Class Encoding

We can encode XOR classes as:

0 and 1

or:

-1 and 1

The encoding changes the symbols.

It does not change the class pattern.

Encoding changes representation of labels, not the underlying XOR relation.

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Why XOR Classes Matter

XOR classes force us to look beyond the rule itself.

The rule is simple:

same -> 0
different -> 1

But the geometry of the classes is not linearly separable in the original 2D input space.

XOR shows that simple logical structure can create difficult geometric structure.

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Final Summary

• XOR has two classes.
• Class 0 means inputs match.
• Class 1 means inputs differ.
• Class 0 points are opposite corners.
• Class 1 points are the other opposite corners.
• The classes alternate across the square.
• This creates a nonlinearly separable pattern.