If f₁ and f₂ are functions from A to R then

1. (f₁ + f₂)(x) = f₁(x) + f₂(x)
2. (f₁ f₂)(x) = f₁(x) f₂(x)

define the sum and product of f₁ and f₂. These new functions are functions from A to R as well.