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:
“Dont you think Ive had enough excitement for one evening, without the additional thrill of a strange man making love to me?”
—John L. Balderston (18991954)
“I would have these good people to recollect, that the laws of this country hold out to foreigners an offer of all that liberty of the press which Americans enjoy, and that, if this liberty be abridged, by whatever means it may be done, the laws and the constitution, and all together, is a mere cheat; a snare to catch the credulous and enthusiastic of every other nation; a downright imposition on the world.”
—William Cobbett (17621835)
“When Hitler attacked the Jews ... I was not a Jew, therefore, I was not concerned. And when Hitler attacked the Catholics, I was not a Catholic, and therefore, I was not concerned. And when Hitler attacked the unions and the industrialists, I was not a member of the unions and I was not concerned. Then, Hitler attacked me and the Protestant churchand there was nobody left to be concerned.”
—Martin Niemller (18921984)