Why does the universal property of the Cartesian product not generalize to addition?

Conditions

• The comparison is between the Cartesian product and addition.

Reasoning, step by step

1. Recall the universal property of the Cartesian product: $(X \times Y)^Z \cong X^Z \times Y^Z$.
2. Consider the analogous property for addition: $(a+b)^2 = a^2 + b^2$.
3. Expand $(a+b)^2$ to get $a^2 + 2ab + b^2$.
4. Observe that $a^2 + 2ab + b^2 \neq a^2 + b^2$ unless $ab=0$.
5. Conclude that the property does not hold for addition.

Example

The speaker uses $(a+b)^2 \neq a^2 + b^2$ to show the property doesn't hold for all operations.

Common misconceptions

• Assuming that all binary operations share the same universal properties.
• Ignoring the cross term in the expansion of $(a+b)^2$.

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