Bridge to Abstract Mathematics: Mathematical Proof and Structures

1.5 COUNTING PROPERTIES OF FINITE SETS (OPTIONAL) 49

ization in mathematics. Example 5 showed that, even when the universal

set is relatively small, the number of cases included by a well-known theo-

rem of set theory is quite large. An abstract proof, of course, establishes

the truth of the theorem in all these special cases and eliminates the need

to count, or otherwise consider, the individual cases. See Exercises 9 and

1 l(c) for other similar examples.

If you are interested in further exposure to this branch of mathematics,

you should refer to texts in areas such as probability and combinatorial

mathematics.

Exercises

1. Find n(A) where A =
   2. The faculty of 34 at a local college has invested its retirement contributions in
   either the stock fund or the money market fund. There are 22 with money in the
   stock fund and 27 with money in the money market fund. How many have invested
   a portion of their money in both funds?


2. An inspection of 63 automobiles available for sale at a local dealership revealed
   that 41 had air conditioning, 31 had cruise control, and 37 had tilt wheel. Also, 27
   have both air conditioning and cruise control, 18 have cruise control and tilt wheel,
   whereas 30 have air conditioning and tilt wheel. Finally, 18 have all three options.
   How many of the cars have none of the options? (Hint: Construct a Venn diagram
   based on three circles. Insert numbers into as many of the eight regions induced
   by the three circles as the preceding data provide for.)


3. Find a formula for n(A u B u C), analogous to the formula (Counting Formula
   
   
   1. for n(A u B). [Hint: Review the reasoning used to justify the formula for n(A u B).
      Your answer should involve n(A), n(B), n(C), and n(various intersections of these
      sets).]
   
   
   
   
   
   
   5. (a) How many committees, with at least one member, can be formed from a club
      with eight members?
      *(b) How many three-person committees can be formed from the club in part (a)?
      five-person commit tees?
      (c) How many ways are there of selecting a president, vice-president, and secretary
      from the club in part (a)?