ISYE 2027 - EXAM 1 QUESTIONS AND
ANSWERS
combinatory - CORRECT ANSWER-branch of discrete mathematics that studies the
arrangement of objects
enumeration - CORRECT ANSWER-part of combinatory that explores the ways we can
count objects with certain properties
product rule - CORRECT ANSWER-to find the number of ways of doing something,
multiply the number of choices available at each stage
sum rule - CORRECT ANSWER-if a task can be done either in one of n1 ways or in one
of n2 ways, where none of the set of n1 ways is the same as any of the set of n2 ways,
then there are n1 + n2 ways to do the task
subtraction rule - CORRECT ANSWER-if a task can be done in either n1 ways or n2
ways, then the number of ways to do the task is equal to n1 + n2 minus the number of
ways to do the task that are common to the two different ways
inclusion-exclusion principle - CORRECT ANSWER-let A, B, C be some finite set:
|A u B| = |A| + |B| - |A n B|
|A u B U C| = |A| + |B| + |C| - |A n B| - |A n C| - |B n C| + |A n B N C|
division rule - CORRECT ANSWER-there are n/d ways to do a task if it can be done
using a procedure that can be carried out in n ways, and for every way w, there exists
exactly d of the n ways that correspond to that way w
pigeonhole principle - CORRECT ANSWER-if k is a positive integer and k + 1 or more
objects are placed into k boxes, then there is at least one box containing two or more of
the objects
permutation - CORRECT ANSWER-set of distinct objects in an ordered arrangement of
these objects
P (n, r) = n (n - 1) (n - 2) ... (n - r + 1) = n! /(n-r)!
combination - CORRECT ANSWER-set of distinct objects in an unordered arrangement
of these objects
C (n, r) = n! / (r! (n-r)!)
, set - CORRECT ANSWER-any well-defined collection of objects
elements - CORRECT ANSWER-individual objects within a set
{}: open/closes a set
x E A: x is member of A
cardinality - CORRECT ANSWER-number of elements in the set
null set - CORRECT ANSWER-set with no elements (ø)
universal set - CORRECT ANSWER-set of all elements under consideration; set of all
possible combinations (omega)
subset - CORRECT ANSWER-A is a subset of B if x E A implies x E B
- all elements of set A are also members of set B
proper subset - CORRECT ANSWER-all members of A are also members of B and
there exists at least one element c such that c E B but c is not an element of A
set union - CORRECT ANSWER-all elements in A, or B, or both (A u B)
set intersection - CORRECT ANSWER-all elements in both A AND B (A n B)
set complement - CORRECT ANSWER-set of all elements not in A (Ā)
set difference - CORRECT ANSWER-set of elements in A that are not in B (A - B)
exhaustive - CORRECT ANSWER-a group of sets is exhaustive of another set if their
UNION is equal to that set
- if A U B = C, then A and B are exhaustive of C
mutually exclusive - CORRECT ANSWER-two sets are mutually exclusive if A N B = ø
(no elements in common)
partition - CORRECT ANSWER-a group of sets partitions another set if they are
mutually exclusive AND exhaustive with respect to that set
- no similar elements & covers all the elements in the set
power set - CORRECT ANSWER-set of all subsets of X (P(X) = {A: A c X}
ANSWERS
combinatory - CORRECT ANSWER-branch of discrete mathematics that studies the
arrangement of objects
enumeration - CORRECT ANSWER-part of combinatory that explores the ways we can
count objects with certain properties
product rule - CORRECT ANSWER-to find the number of ways of doing something,
multiply the number of choices available at each stage
sum rule - CORRECT ANSWER-if a task can be done either in one of n1 ways or in one
of n2 ways, where none of the set of n1 ways is the same as any of the set of n2 ways,
then there are n1 + n2 ways to do the task
subtraction rule - CORRECT ANSWER-if a task can be done in either n1 ways or n2
ways, then the number of ways to do the task is equal to n1 + n2 minus the number of
ways to do the task that are common to the two different ways
inclusion-exclusion principle - CORRECT ANSWER-let A, B, C be some finite set:
|A u B| = |A| + |B| - |A n B|
|A u B U C| = |A| + |B| + |C| - |A n B| - |A n C| - |B n C| + |A n B N C|
division rule - CORRECT ANSWER-there are n/d ways to do a task if it can be done
using a procedure that can be carried out in n ways, and for every way w, there exists
exactly d of the n ways that correspond to that way w
pigeonhole principle - CORRECT ANSWER-if k is a positive integer and k + 1 or more
objects are placed into k boxes, then there is at least one box containing two or more of
the objects
permutation - CORRECT ANSWER-set of distinct objects in an ordered arrangement of
these objects
P (n, r) = n (n - 1) (n - 2) ... (n - r + 1) = n! /(n-r)!
combination - CORRECT ANSWER-set of distinct objects in an unordered arrangement
of these objects
C (n, r) = n! / (r! (n-r)!)
, set - CORRECT ANSWER-any well-defined collection of objects
elements - CORRECT ANSWER-individual objects within a set
{}: open/closes a set
x E A: x is member of A
cardinality - CORRECT ANSWER-number of elements in the set
null set - CORRECT ANSWER-set with no elements (ø)
universal set - CORRECT ANSWER-set of all elements under consideration; set of all
possible combinations (omega)
subset - CORRECT ANSWER-A is a subset of B if x E A implies x E B
- all elements of set A are also members of set B
proper subset - CORRECT ANSWER-all members of A are also members of B and
there exists at least one element c such that c E B but c is not an element of A
set union - CORRECT ANSWER-all elements in A, or B, or both (A u B)
set intersection - CORRECT ANSWER-all elements in both A AND B (A n B)
set complement - CORRECT ANSWER-set of all elements not in A (Ā)
set difference - CORRECT ANSWER-set of elements in A that are not in B (A - B)
exhaustive - CORRECT ANSWER-a group of sets is exhaustive of another set if their
UNION is equal to that set
- if A U B = C, then A and B are exhaustive of C
mutually exclusive - CORRECT ANSWER-two sets are mutually exclusive if A N B = ø
(no elements in common)
partition - CORRECT ANSWER-a group of sets partitions another set if they are
mutually exclusive AND exhaustive with respect to that set
- no similar elements & covers all the elements in the set
power set - CORRECT ANSWER-set of all subsets of X (P(X) = {A: A c X}
