Posts tagged with complement

Yesterday’s #SB5 conflict is an opportunity to talk about the algebra of sets. (Commonly understood through Venn diagrams.)

The algebra of sets is the way ∪, ∩, , and some composite operations work. These words are useful for reminding yourself to logically separate things that are logically separate.

The relative complement of A (left circle) in B (right circle):
$A^c \cap B~~~~=~~~~B \smallsetminus A$

For example:

• Not all feminists are women.

`Feminist ∩ ∁{Woman}`
• Not all women are pro-choice.

`∁{Pro-Choice} ∩ Woman`

Sets can overlap in different ways.

## Complement

Here’s an inside-out thought: The air around us is a 3-manifold with 3-holes where solid objects are, and the 2-boundary is the ground. Or if you think of all the sky, it’s a spherical 3-shell (with one 3-hole, the Earth) floating in empty space.

I wish I could draw what I’m thinking of. Something like this.

From a child’s-eye view, “the air” is the complement of everything that’s “actually there” (solid or liquid things).

$\large \dpi{200} \bg_white \text{air} = \complement (\text{what's there})$

$\large \dpi{150} \bg_white \{\text{what's there} \} \equiv \{\text{solids}\} \cup \{\text{liquids} \}$

You could correct that child by bringing up outer space (another 2-boundary on the air), or the fact that air is made of particles as well.

$\large \dpi{200} \bg_white \partial \; \text{air} = \partial \{\text{outer space}\} \cup \partial \{\text{the ground}\}$

But wouldn’t you rather be sitting in the middle of a field imagining yourself, the trees, the grass, the clouds, the birds, and the wood chips being cut out by a Photoshop lasso?