Schaum's Outline of Discrete Mathematics, Third Edition (Schaum's Outlines)

CHAP. 1] SET THEORY 5

Fig. 1-3

Recall that setsAandBare said to bedisjointornonintersectingif they have no elements in common or,
using the definition of intersection, ifA∩B=∅, the empty set. Suppose

S=A∪B and A∩B=∅

ThenSis called thedisjoint unionofAandB.

EXAMPLE 1.4

(a) Let A={1, 2, 3, 4},B={3, 4, 5, 6, 7},C={2, 3, 8, 9}. Then

A∪B={ 1 , 2 , 3 , 4 , 5 , 6 , 7 },A∪C={ 1 , 2 , 3 , 4 , 8 , 9 },B∪C={ 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 },

A∩B={ 3 , 4 },A∩C={ 2 , 3 },B∩C={ 3 }.

(b) LetUbethe set of students at a university, and letMdenote the set of male students and letFdenote the set

of female students. TheUis the disjoint union ofMofF; that is,

U=M∪F and M∩F=∅

This comes from the fact that every student inUis either inMor inF, and clearly no student belongs to

bothMandF, that is,MandFare disjoint.

The following properties of union and intersection should be noted.

Property 1: Every elementxinA∩Bbelongs to bothAandB; hencexbelongs toAandxbelongs toB. Thus
A∩Bis a subset ofAand ofB; namely

A∩B⊆A and A∩B⊆B

Property2: An elementxbelongs to the unionA∪Bifxbelongs toAorxbelongs toB; hence every element
inAbelongs toA∪B, and every element inBbelongs toA∪B. That is,

A⊆A∪B and B⊆A∪B

We state the above results formally:

Theorem 1.3: For any setsAandB, we have:

(i)A∩B⊆A⊆A∪Band (ii)A∩B⊆B⊆A∪B.
The operation of set inclusion is closely related to the operations of union and intersection, as shown by the
following theorem.

Theorem 1.4: The following are equivalent:A⊆B, A∩B=A, A∪B=B.

This theorem is proved in Problem 1.8. Other equivalent conditions to are given in Problem 1.31.