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

Famous quotes containing the words additional, laws and/or unions:

    The world will never be long without some good reason to hate the unhappy; their real faults are immediately detected, and if those are not sufficient to sink them into infamy, an additional weight of calumny will be superadded.
    Samuel Johnson (1709–1784)

    The counting-room maxims liberally expounded are laws of the Universe. The merchant’s economy is a coarse symbol of the soul’s economy. It is, to spend for power, and not for pleasure.
    Ralph Waldo Emerson (1803–1882)

    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)