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
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