# Free Disposal: the production set less the vector space of positive reals

In Microeconomic Theory (3rd Edition), Mas-Collel, Whinston and Green state that the property of free disposal implies the following:

$$Y - \mathbb{R}^L_+ \subset Y$$

Y: Production set

y: Production vector

L: Amount of commodities considered

Why does this formula means that the extra amount of inputs or outputs can be disposed of or eliminated at no cost?

So the way that $$Y$$ is defined is such that negative values are inputs and positive values are outputs. For example, if I used 3 units of labor, and 2 of capital to produce 10 pencils, that will be reflected as a single vector in $$\mathbb{R}^3$$ equal to $$[-3;-2;10]\in Y$$.
Notice that subtracting 1 unit to either the first or the second element of that vector will mean that I used more inputs to produce the same output, for example, $$[-3;-3;-10]$$. Similarly, if I subtract one unit to the third element, it would mean that I produced less output with the same inputs. For example $$[-3;-2;9]$$.
Therefore you can conclude that a production set, $$Y$$, satisfies free disposal if and only if $$Y-\mathbb{R}^L_+\subset Y$$.