1.5 COUNTING PROPERTIES OF FINITE SETS (OPTIONAL) 51
Figure 1.12
*
a
b
C
(c) Let U = (1,2,3,... ,9, 10). Use the formula from part (a) of this exercise to
calculate the number of specific cases encompassed by Conjecture 5, Article 1.3.
IfX E Y,then(YnZ)uX= Yn(ZuX).
abc
- Suppose we wish to construct a multiplication table, as in Figure 1.12a, based
on ,a three-element set S = (a, b, c). We define the "product" x y of two elements
in S by entering a symbol in the box corresponding to the row of x and the column
of y. Thus Figure 1.12b indicates that a b = c. Assuming that only the symbols
a, b, and c may be used to fill each of the empty boxes in Figure 1.12b:
(a) How many possible ways are there to construct such a table?
(b) How many such tables can be constructed if must be bbcommutative"; that
is, if x y = y * x for all elements x, y E S?
(c) How many tables are possible if none of the symbols a, b, and c can be used
more than once in any row?
(d) How many tables are possible if no symbol can be used more than once in
any row and once in any column?
(a)
