Discrete Mathematics: Elementary and Beyond

22 1. Let’s Count!

Review Exercises

1.8.8In how many ways can you seat 12 people at two round tables with 6
places each? Think of possible ways of defining when two seatings are different,
and find the answer for each.

1.8.9Name sets with cardinality (a) 365, (b) 12, (c) 7, (d) 11.5, (e) 0, (f) 1024.

1.8.10List all subsets of{a, b, c, d, e}containing{a, e}but not containingc.

1.8.11We have not written up all subset relations between various sets of num-
bers; for example,
⊆
is also true. How many such relations can you find
between the sets∅,,+,,,
?

1.8.12What is the intersection of

(a) the set of positive integers whose last digit is 3, and the set of even numbers;

(b) the set of integers divisible by 5 and the set of even integers?

1.8.13LetA={a, b, c, d, e}andB={c, d, e}. List all subsets ofAwhose
intersection withBhas 1 element.

1.8.14Three sets have 5, 10, and 15 elements, respectively. How many elements
can their union and their intersection have?

1.8.15What is the symmetric difference ofAandA?

1.8.16Form the symmetric difference ofAandBto get a setC. Form the
symmetric difference ofAandC. What set do you get?

1.8.17LetA, B, Cbe three sets and assume thatAis a subset ofC. Prove that

A∪(B∩C)=(A∪B)∩C.

Show by an example that the condition thatAis a subset ofCcannot be omitted.

1.8.18What is the differenceA\Bif

(a)Ais the set of primes andBis the set of odd integers?

(b) Ais the set of nonnegative real numbers andBis the set of nonpositive

real numbers?

1.8.19Prove that for any three setsA, B, C,

((A\B)∪(B\A))∩C=((A∩C)∪(B∩C))\(A∩B∩C).