The identity laws: A ∪ ∅ = A and A ∩ U = A¶

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

One line: Adding nothing changes nothing, and keeping everything keeps A: ∅ is the identity for ∪ and U for ∩, the way 0 is for + and 1 for ×.

The law¶

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

Why it is true¶

For any x, "x ∈ A or x ∈ ∅" is just "x ∈ A", since x ∈ ∅ is always false; and "x ∈ A and x ∈ U" is just "x ∈ A", since x ∈ U is always true inside the universe. In logic, or false and and true both leave p alone. 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¶

The identity elements are what make the subsets of U a monoid under ∪ and again under ∩, and in code a | set() and a & U are the no-ops. The trap is the second: A ∩ U = A only because U was fixed first; with a bigger U the complement changes, though this law does not.

What the program prints¶

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

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

3. THE SAME LAW IN LOGIC: p ∨ false = p, p ∧ true = 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   T     T

Po polsku, w skrócie¶

Prawa identyczności: suma ze zbiorem pustym i przekrój z uniwersum nie zmieniają A, tak jak dodanie 0 i pomnożenie przez 1 nie zmieniają liczby. W logice to „lub fałsz" i „i prawda".

Auf Deutsch: Stichwörter¶

Neutrale Elemente: A ∪ ∅ = A und A ∩ U = A.

Stichwörter: neutrales Element (identity), leere Menge, Grundmenge.