The idempotent laws: A ∪ A = A and A ∩ A = A¶

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

One line: Doing it twice is doing it once: a set joined or met with itself is itself, which numbers under + and × do not do, and which is the reason sets have no "2A".

The law¶

Sets	Logic	Python
A ∪ A = A, A ∩ A = A	p ∨ p = p, p ∧ p = p	`a \| a == a and a & a == a`

Why it is true¶

"x ∈ A or x ∈ A" and "x ∈ A and x ∈ A" each say "x ∈ A". A repeated member is not a second member, which is extensionality again: {1, 1} = {1}. 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¶

Idempotence is what makes a set operation safe to repeat: a database UNION applied twice, a set.update run twice, a tag added twice, all leave one copy. Numbers fail it (a + a = 2a), and so do multisets, where [2] + [2] = [2, 2]; max and min on numbers pass it, which is why max and min behave like ∪ and ∩.

What the program prints¶

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

1. THE LAW: A ∪ A = A, A ∩ 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} = A
   A ∩ A = {1, 2} = A

3. THE SAME LAW IN LOGIC: p ∨ p = p, p ∧ p = p
   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   F     F

Po polsku, w skrócie¶

Prawa idempotentności: suma i przekrój zbioru z samym sobą to ten sam zbiór, bo powtórzony element to nie drugi element. Liczby tego nie mają (a + a = 2a), max i min mają.

Auf Deutsch: Stichwörter¶

Idempotenzgesetze: A ∪ A = A und A ∩ A = A.

Stichwörter: Idempotenz, Vereinigung, Schnitt, keine Wiederholung in Mengen.