Algebra of Sets - Some Additional Laws For Unions and Intersections

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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