Algebraic Type System

def. Algebraic Type System uses both:

• Product Types: stuct Icon: {shape: Shape, x_pos: Int, y_pos: Int}
• Sum Types: enum Shape: Square | Circle | Triangle Intuition. It’s called algebraic because:

$$\begin{array}{rlr}\text{Icon}& \in \text{Shape}\times \text{Int}\times \text{Int}& \text{Product Type}\\ \text{Shape}& =\{\text{Square},\text{Circle},\text{Triangle}\}& \text{Sum Type}\end{array}$$

Thus looking at its cardinalities:

$$\begin{array}{rl}|\text{Icon}|& =|\text{Shape}|\cdot |\text{Int}|\cdot |\text{Int}|\\ & =(|\text{Square}|+|\text{Circle}|+|\text{Triangle}|)\cdot |\text{Int}|\cdot |\text{Int}|\\ & =3\cdot |\mathbb{N}|\cdot |\mathbb{N}|\end{array}$$