Mathematical Methods for Physics and Engineering : A Comprehensive Guide

30.1 VENN DIAGRAMS

gets beyond three it becomes impossible to draw a simple two-dimensional Venn

diagram, but this does not change the results.

The 2nregions will break down inton+ 1 types, with the numbers of each type

as follows§

no events, nC 0 =1;

one event but no intersections, nC 1 =n;

two-fold intersections, nC 2 =^12 n(n−1);

three-fold intersections, nC 3 =3!^1 n(n−1)(n−2);

..

.

ann-fold intersection, nCn=1.

That this makes a total of 2ncan be checked by considering the binomial

expansion

2 n=(1+1)n=1+n+^12 n(n−1) +···+1.

Using Venn diagrams, it is straightforward to show that the operations∩and

∪obey the following algebraic laws:

commutativity, A∩B=B∩A, A∪B=B∪A;

associativity, (A∩B)∩C=A∩(B∩C), (A∪B)∪C=A∪(B∪C);

distributivity, A∩(B∪C)=(A∩B)∪(A∩C),

A∪(B∩C)=(A∪B)∩(A∪C);

idempotency, A∩A=A, A∪A=A.

Show that(i)A∪(A∩B)=A∩(A∪B)=A,(ii) (A−B)∪(A∩B)=A.

(i) Using the distributivity and idempotency laws above, we see that

A∪(A∩B)=(A∪A)∩(A∪B)=A∩(A∪B).

By sketching a Venn diagram it is immediately clear that both expressions are equal to
A. Nevertheless, we here proceed in a more formal manner in order to deduce this result
algebraically. Let us begin by writing

X=A∪(A∩B)=A∩(A∪B), (30.4)

from which we want to deduce a simpler expression for the eventX. Using the first equality
in (30.4) and the algebraic laws for∩and∪,wemaywrite

A∩X=A∩[A∪(A∩B)]

=(A∩A)∪[A∩(A∩B)]

=A∪(A∩B)=X.

§The symbolsnCi,fori=0, 1 , 2 ,...,n, are a convenient notation for combinations; they and their

properties are discussed in chapter 1.