8 1. Let’s Count!
We get this simply by interchanging “union” and “intersection” in (1.1).
(This identity can be proved just like (1.1); see Exercise 1.2.16.) This second
distributivity is something that has no analogue for numbers: In general,
a+bc =(a+b)(a+c)for three numbersa, b, c.
There are other remarkable identities involving union, intersection, and
also the two kinds of differences. These are useful, but not very deep: They
reflect simple logic. So we don’t list them here, but state several of these
below in the exercises.
1.2.1Name sets whose elements are (a) buildings, (b) people, (c) students, (d)
trees, (e) numbers, (f) points.
1.2.2What are the elements of the following sets: (a) army, (b) mankind, (c)
library, (d) the animal kingdom?
1.2.3Name sets having cardinality (a) 52, (b) 13, (c) 32, (d) 100, (e) 90, (f)
2,000,000.
1.2.4What are the elements of the following (admittedly peculiar) set:
{Alice,{ 1 }}?
1.2.5Is an “element of a set” a special case of a “subset of a set”?1.2.6List all subsets of{ 0 , 1 , 3 }. How many do you get?1.2.7Define at least three sets of which{Alice, Diane, Eve}is a subset.1.2.8List all subsets of{a, b, c, d, e}, containingabut not containingb.1.2.9Define a set of which both{ 1 , 3 , 4 }and{ 0 , 3 , 5 }are subsets. Find such a
set with the smallest possible number of elements.
1.2.10 (a) Which set would you call the union of{a, b, c},{a, b, d}and
{b, c, d, e}?
(b) Find the union of the first two sets, and then the union of this with the
third. Also, find the union of the last two sets, and then the union of this
with the first set. Try to formulate what you have observed.
(c) Give a definition of the union of more than two sets.1.2.11Explain the connection between the notion of the union of sets and
Exercise 1.2.9.
1.2.12We form the union of a set with 5 elements and a set with 9 elements.
Which of the following numbers can we get as the cardinality of the union: 4, 6,
9, 10, 14, 20?