Appendix C Worked-Out Solutions to Exercises: Chapters 21 to
it’s not the same thing as the null set. It can be the sole element of another set, namely
{{∅}}. By now, you should be able to sense where this is leading:
∅
{∅}
{{∅}}
{{{∅}}}
{{{{∅}}}}
{{{{{∅}}}}}
↓
and so on, forever
You started with nothing, and you have turned it into an infinite number of mathematical
objects! A variation of this idea is handy for “building” numbers, as you’ll see in Chap. 3.
- In Fig. 2-6, set P, represented by the small dark-shaded triangle, is common to (that is,
shared by) sets A and C. Therefore, region P represents the intersection of sets A and C.
You can write this as
P=A∩C
Set Q, represented by the small, dark-shaded, irregular four-sided figure, is fully shared
by sets B and D.Therefore, region Q represents the intersection of sets B and D, which
you can write as
Q=B∩D
- Whenever two regions are entirely separate, then the sets they represent are disjoint, and
the intersection of those sets is the null set. You can see from Fig. 2-6 that the only pairs
of regions that don’t overlap are A and D,B and C, and C and D. Therefore, the only
null-set intersection pairs are
A∩D=∅
B∩C=∅
C∩D=∅
- The universal set (call it U) is a subset of itself. That’s trivial, because any set is a subset
of itself. But U is not a proper subset of itself. Remember, U is the set of all entities,
real or imaginary. If U were a proper subset of itself, then there would be some entity
that did not belong to U. That’s impossible; it contradicts the very definition of U!
This little argument is an example of a tactic called reductio ad absurdum (Latin for
“reduction to absurdity”) that mathematicians have used for thousands of years to prove
or disprove “slippery statements.” It can work well in a courtroom, too.
- There are plenty of examples that will work here. The set of all even whole numbers,
Weven, is a proper subset of the set of all whole numbers, W. Both of these sets have
