The complement laws: A ∪ A′ = U and A ∩ A′ = ∅¶

Level: 101 · one law of the algebra of sets, on its own page

One line: Every member of the universe is in A or out of it, and none is both: the law of the excluded middle and the law of non-contradiction, written for sets.

The law¶

Sets	Logic	Python
A ∪ A′ = U, A ∩ A′ = ∅	p ∨ ¬p = true, p ∧ ¬p = false	`a \| (U - a) == U and a & (U - a) == set()`

Why it is true¶

For each x in U, "x ∈ A or x ∉ A" is true, since one of the two must hold, and "x ∈ A and x ∉ A" is false, since both cannot. So the union of A with its complement is all of U and the intersection is empty. The complement needs U fixed in advance: A′ means U ∖ A, and the parent page says why. A law of sets is decided one member at a time, so checking it on every subset of a four-element universe, every pattern of membership there is, proves it; the program does that and prints the matching truth table.

Where it is used, and the trap¶

This is the law behind every case split: to prove something for all x, prove it for x ∈ A and for x ∉ A, and the two cases cover everything and overlap nowhere. It is also the one law a fuzzy or three-valued logic gives up, as day or night ↗ shows: twilight is in "day" and in "not day" a little each.

What the program prints¶

Verified output of law_complement.py — regenerated by tools/run_examples.py, never hand-typed.

1. THE LAW: A ∪ A′ = U, A ∩ A′ = ∅
   checked on all 16 choices of 1 subset(s) of U = {1, 2, 3, 4}: True

2. ONE INSTANCE, WORKED
   A = {1, 2}, B = {2, 3}, C = {3, 4}, A' = {3, 4}, B' = {1, 4}
   A ∪ A' = {1, 2, 3, 4} = U
   A ∩ A' = ∅ = ∅

3. THE SAME LAW IN LOGIC: p ∨ ¬p = true, p ∧ ¬p = false
   every row of the truth table agrees, which is why the member-by-member check is a proof:
   p   left  right
   T   T     T
   F   T     T

Po polsku, w skrócie¶

Prawa dopełnienia: każdy element uniwersum należy do A albo do A′, i żaden do obu; to prawo wyłączonego środka i prawo niesprzeczności zapisane dla zbiorów. Dopełnienie wymaga ustalonego wcześniej U.

Auf Deutsch: Stichwörter¶

Komplementgesetze: A ∪ A′ = U und A ∩ A′ = ∅.

Stichwörter: Komplement, Grundmenge U, leere Menge, Partition in A und A′.