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:
“The mere existence of an additional child or children in the family could signify Less. Less time alone with parents. Less attention for hurts and disappointments. Less approval for accomplishments. . . . No wonder children struggle so fiercely to be first or best. No wonder they mobilize all their energy to have more or most. Or better still, all.”
—Adele Faber (20th century)
“The great end of all human industry is the attainment of happiness. For this were arts invented, sciences cultivated, laws ordained, and societies modelled, by the most profound wisdom of patriots and legislators. Even the lonely savage, who lies exposed to the inclemency of the elements and the fury of wild beasts, forgets not, for a moment, this grand object of his being.”
—David Hume (1711–1776)
“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)