The absorption laws: A ∪ (A ∩ B) = A and A ∩ (A ∪ B) = A¶

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

One line: A ∩ B already lies inside A, so joining it adds nothing; A ∪ B already contains A, so meeting with it removes nothing: B is absorbed and disappears.

The law¶

Sets	Logic	Python
A ∪ (A ∩ B) = A, A ∩ (A ∪ B) = A	p ∨ (p ∧ q) = p, p ∧ (p ∨ q) = p	`a \| (a & b) == a`

Why it is true¶

If x ∈ A then x is in both sides of each law; if x ∉ A then x ∉ A ∩ B, so x is in neither side of the first, and x ∉ A ∩ (A ∪ B), so in neither side of the second. B never gets a say. In logic, "p or (p and q)" is true exactly when p is. 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¶

Absorption is the rule a simplifier uses to delete redundant conditions: a or (a and b) is a, and a query that filters by A and then by "A or B" filtered by A. It is also the law that defines a lattice, the structure ∪ and ∩ share with max and min, gcd and lcm.

What the program prints¶

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

1. THE LAW: A ∪ (A ∩ B) = A, A ∩ (A ∪ B) = A
   checked on all 256 choices of 2 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 ∩ B) = A ∪ {2} = {1, 2} = A
   A ∩ (A ∪ B) = A ∩ {1, 2, 3} = {1, 2} = A

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

Po polsku, w skrócie¶

Prawa absorpcji: A ∩ B leży wewnątrz A, więc suma nic nie dodaje; A ∪ B zawiera A, więc przekrój nic nie zabiera; B znika. Tak upraszcza się warunki: a or (a and b) to a.

Auf Deutsch: Stichwörter¶

Absorptionsgesetze: A ∪ (A ∩ B) = A und A ∩ (A ∪ B) = A.

Stichwörter: Absorptionsgesetz, Vereinigung, Schnitt, Verband (lattice).