c H a p t e R 4 review
teRminologY
addition principle (p. 254)
Bayes’ theorem (p. 309)
Bernoulli trial (p. 313)
Bernoulli experiment (p. 313)
binary operation (p. 229)
binomial coefficient (p. 297)
binomial distribution (p.313)
binomial theorem (p. 296)
Cantor’s diagonalization method
(p. 237)
cardinality of a set (p. 236)
Cartesian product (cross product)
of sets (p. 233)
closed set under an operation
(p. 229)
combination (p. 274)
combinatorial proof (p. 296)
combinatorics (p. 252)
complement of a set (p. 232)
conditional probability (p. 307)
countable set (p. 236)
decision tree (p. 257)
denumerable set (p. 236)
disjoint sets (p. 232)
dual of a set identity (p. 235)
empty set (p. 224)
equal sets (p. 223)
event (p. 302)
expected value (p. 310)
finite set (p. 223)
independent events (p. 307)
infinite set (p. 223)
intersection of sets (p. 231)
lexicographical ordering (p. 281)
linearity of expected value
(p. 312)
multiplication principle (p. 253)
n factorial (p. 272)
null set (p. 224)
ordered pair (p. 228)
Pascal’s formula (p. 295)
Pascal’s triangle (p. 294)
permutation (p. 272)
pigeonhole principle (p. 269)
power set (p. 227)
principle of inclusion and
exclusion (p. 267)
probability axioms (p. 304)
probability distribution (p. 305)
probability of an event (p. 302)
probability of an event E (p. 305)
proper subset (p. 225)
random variable (p. 310)
sample space (p. 302)
set difference (p. 232)
subset (p. 224)
unary operation (p. 230)
uncountable set (p. 236)
union of sets (p. 231)
universal set (p. 231)
universe of discourse (p. 231)
weighted average (p. 310)
well-defined operation (p. 229)
Self-teSt
Answer the following true−false questions.
Section 4.1
1. The empty set is a proper subset of every set.
2. If A and B are disjoint sets, then (A − B) c (B − A)
= A c B.
3. If a set has n elements, then its power set has 2
n
elements.
4. If a binary operation + on a set S is well-defined,
then x + y [ S for all x and y in S.
5. Cantor’s diagonalization method is a way to prove
that certain sets are denumerable.
Section 4.2
1. According to the multiplication principle, the number
of outcomes for a sequence of tasks is the product of
the number of outcomes for each separate task.
2. The addition principle finds the total number of
branches of a decision tree.
3. The addition principle requires the tasks at hand to
have disjoint sets of outcomes.
4. The multiplication principle says that the number of
elements in A × B equals the number of elements in
A times the number of elements in B.
5. Any problem that requires a decision tree for its
solution cannot be solved by the multiplication
principle.
Section 4.3
1. The principle of inclusion and exclusion requires
that A and B be disjoint sets in order to find the
number of elements in A c B.
2. The principle of inclusion and exclusion applied
to two sets says that the number of elements in
the union minus the number of elements in the
intersection is the sum of the number of elements
in each set.
Chapter 4 Review
323
teRminologY
addition principle (p. 254)
Bayes’ theorem (p. 309)
Bernoulli trial (p. 313)
Bernoulli experiment (p. 313)
binary operation (p. 229)
binomial coefficient (p. 297)
binomial distribution (p.313)
binomial theorem (p. 296)
Cantor’s diagonalization method
(p. 237)
cardinality of a set (p. 236)
Cartesian product (cross product)
of sets (p. 233)
closed set under an operation
(p. 229)
combination (p. 274)
combinatorial proof (p. 296)
combinatorics (p. 252)
complement of a set (p. 232)
conditional probability (p. 307)
countable set (p. 236)
decision tree (p. 257)
denumerable set (p. 236)
disjoint sets (p. 232)
dual of a set identity (p. 235)
empty set (p. 224)
equal sets (p. 223)
event (p. 302)
expected value (p. 310)
finite set (p. 223)
independent events (p. 307)
infinite set (p. 223)
intersection of sets (p. 231)
lexicographical ordering (p. 281)
linearity of expected value
(p. 312)
multiplication principle (p. 253)
n factorial (p. 272)
null set (p. 224)
ordered pair (p. 228)
Pascal’s formula (p. 295)
Pascal’s triangle (p. 294)
permutation (p. 272)
pigeonhole principle (p. 269)
power set (p. 227)
principle of inclusion and
exclusion (p. 267)
probability axioms (p. 304)
probability distribution (p. 305)
probability of an event (p. 302)
probability of an event E (p. 305)
proper subset (p. 225)
random variable (p. 310)
sample space (p. 302)
set difference (p. 232)
subset (p. 224)
unary operation (p. 230)
uncountable set (p. 236)
union of sets (p. 231)
universal set (p. 231)
universe of discourse (p. 231)
weighted average (p. 310)
well-defined operation (p. 229)
Self-teSt
Answer the following true−false questions.
Section 4.1
1. The empty set is a proper subset of every set.
2. If A and B are disjoint sets, then (A − B) c (B − A)
= A c B.
3. If a set has n elements, then its power set has 2
n
elements.
4. If a binary operation + on a set S is well-defined,
then x + y [ S for all x and y in S.
5. Cantor’s diagonalization method is a way to prove
that certain sets are denumerable.
Section 4.2
1. According to the multiplication principle, the number
of outcomes for a sequence of tasks is the product of
the number of outcomes for each separate task.
2. The addition principle finds the total number of
branches of a decision tree.
3. The addition principle requires the tasks at hand to
have disjoint sets of outcomes.
4. The multiplication principle says that the number of
elements in A × B equals the number of elements in
A times the number of elements in B.
5. Any problem that requires a decision tree for its
solution cannot be solved by the multiplication
principle.
Section 4.3
1. The principle of inclusion and exclusion requires
that A and B be disjoint sets in order to find the
number of elements in A c B.
2. The principle of inclusion and exclusion applied
to two sets says that the number of elements in
the union minus the number of elements in the
intersection is the sum of the number of elements
in each set.
Chapter 4 Review
323