Some Additional Laws For Unions and Intersections
The following proposition states six more important laws of set algebra, involving unions and intersections.
PROPOSITION 3: For any subsets A and B of a universal set U, the following identities hold:
- idempotent laws:
-
- domination laws:
-
- absorption laws:
-
As noted above each of the laws stated in proposition 3, can be derived from the five fundamental pairs of laws stated in proposition 1 and proposition 2. As an illustration, a proof is given below for the idempotent law for union.
Proof:
| by the identity law of intersection | ||
| by the complement law for union | ||
| by the distributive law of union over intersection | ||
| by the complement law for intersection | ||
| by the identity law for union |
The following proof illustrates that the dual of the above proof is the proof of the dual of the idempotent law for union, namely the idempotent law for intersection.
Proof:
| by the identity law for union | ||
| by the complement law for intersection | ||
| by the distributive law of intersection over union | ||
| by the complement law for union | ||
| by the identity law for intersection |
Intersection can be expressed in terms of union and set difference :
Read more about this topic: Algebra Of Sets
Famous quotes containing the words additional, laws and/or unions:
“Don’t you think I’ve had enough excitement for one evening, without the additional thrill of a strange man making love to me?”
—John L. Balderston (1899–1954)
“We are told to maintain constitutions because they are constitutions, and what is laid down in those constitutions?... Certain great fundamental ideas of right are common to the world, and ... all laws of man’s making which trample on these ideas, are null and void—wrong to obey, right to disobey. The Constitution of the United States recognizes human slavery; and makes the souls of men articles of purchase and of sale.”
—Anna Elizabeth Dickinson (1842–1932)
“The newly-formed clothing unions are ready to welcome her; but woman shrinks back from organization, Heaven knows why! It is perhaps because in organization one find the truest freedom, and woman has been a slave too long to know what freedom means.”
—Katharine Pearson Woods (1853–1923)