Cartesian Product

The Cartesian product of two Set $A$ and $B$ is the set of all ordered pairs $(a,b)$ where $a\in A$ and $b\in B$.

Notation: $A\times B=\{(a,b)\mid a\in A,b\in B\}$

Interpretation as a function:

• The Cartesian product can be seen as a Function: $\times :\mathrm{Set}\times \mathrm{Set}\to \mathrm{Sets}\mathrm{of}\mathrm{Tuples}$
• Meaning: it takes two sets as input and returns a set of tuples as output.

Example:

• $A=\{1,2\},B=\{x,y\}$
• $A\times B=\{(1,x),(1,y),(2,x),(2,y)\}$

Cartesian Product of Multiple Sets: $A_1\times A_2\times \cdots \times A_n=\{(a_1,a_2,\times ,a_n)\mid a_i\in A_i\}$

• Example: $A=\{1,2\},B=\{x,y\},C=\{\alpha \}$
• $A\times B\times C=\{(1,x,\alpha ),(1,y,\alpha ),(2,x,\alpha ),(2,y,\alpha )\}$

Properties:

1. Order matters: $(a,b)\ne (b,a)$ in general
2. Non-commutative: $A\times B\ne B\times A$
3. Associative (up to Isomorphism): $(A\times B)\times C\cong A\times (B\times C)$
4. Identity Element: $A\times \varnothing =\varnothing \times A=\varnothing$
5. Cardinality: $|A\times B|=|A|\cdot |B|$