CS 2050 Full Class
Product Rule - ANS-Way to count sequences
Let A_1, ..., A_n be finite sets. Then |A_1 ✱ ... ✱ A_n | = |A_1| ✱ ... ✱ |A_n|
Alphabet - ANS-Set of characters that are used to form strings; represented by Σ
Σ^n - ANS-Set of all strings of length n that come from set Σ
Sum Rule - ANS-probability of events happening can be calculated by summing
individual probabilities; usually used on mutually exclusive events (both can't happen at
same time)
Generalized Product Rule - ANS-says that in selecting an item from a set, if the number
of choices at each step does not depend on previous choices made, then the number of
items in the set is the product of the number of choices in each step.
Pigeonhole Principle - ANS-if n items are put into m containers, with n > m, then at least
one container must contain more than one item
Generalized Pigeonhole Principle - ANS-If N objects are placed into k boxes, then there
is at least one box containing at least ⌈N/k⌉ objects.
Converse of the Generalized Pigeonhole Principle - ANS-Suppose that a function f
maps a set of n elements to a target set with k elements, where n and k are positive
integers. In order to guarantee that there is an element y in the target to which f maps at
least b elements from the domain, then n must be at least k(b - 1) + 1.
Subtraction Rule - ANS-If a task can be done in either n1 or n2 ways, then the number
of ways to do the task is n1 + n2 minus number of overlapping ways
r-permutation - ANS-Sequence of r items with no repetition, all taken from the same set,
where order matters
r-permutations from set with n elements equals P(n, r) = n!/(n-r)!
,Permutation - ANS-Sequence that contains each element of a finite set exactly once
where order does matter; number of permutations of finite set with n elements is P(n, n)
= n!
r-subsets/r-combinations - ANS-Subsets of size r, where the order does not matter
number of r-subsets from set of n = C(n, r) = P(n, r)/r! = n!/(r!(n-r)!)
Counting Subsets Notation - ANS-C(n, r) equal to (n r) equals n!/(r!(n-r)!); can be read
as "n choose r"
Choosing P(n, r) or C(n, r) - ANS-If order of elements is important, choose P; if not or if
it says "how many ways is there to select", choose C
Counting by Complement - ANS-Technique for counting number of elements in set S
that have a property by counting number without property then subtracting total
elements - those without
Permutations with Repetition - ANS-n^r
Combinations with Repetition - ANS-(n+r-1)!/r!(n-1)!
Permutations with Indistinguishable Objects - ANS-Where n is the total number of
objects, n1 is # of type 1 objects, n2 is # of type 2, etc:
n!/(n1!)(n2!)...(nk!)
Multisets - ANS-Collections with multiple instances of same item where order doesn't
matter
Counting Multisets - ANS-Number of ways to select n objects from set of m varieties is
n+m-1 choose m-1
Definition of Division - ANS-x divides y (x|y) means that there is an integer k such that y
= kx
Linear Combination - ANS-Sum of multiples of two numbers (sx + ty)
Divisibility & Linear Combinations - ANS-If x|y and x|z, then x|(sy + tz)
, Addition and Multiplication mod m - ANS-Operations defined by adding/multiplying two
numbers from a set then applying mod m to the result
Ring - ANS-Set {0, 1, ..., m-1} with add/multiply mod m defines a closed mathematical
system with m elements
Congruent Integers - ANS-Integers that are equivalent in form of x mod m = y mod m;
denoted as x ≡ y (mod m)
Arithmetic Operations mod m - ANS-x mod m + y mod m = [x+y] mod m
x mod m * y mod m = [xy] mod m
Prime Numbers - ANS-Integer greater than 1 with only factors being 1 and itself
Composite Numbers - ANS-Positive integer with factors other than 1 or itself
Prime Factorization - ANS-Way of expressing a positive >1 integer as a product of its
primes
E.g. 112 = 2^4 * 7
Multiplicity of Prime Factor - ANS-Number of times that prime factor appears in the
prime factorization
E.g. in 112 = 2^4 * 7, 2 has a multiplicity of 4
Greatest Common Divisor (gcd) - ANS-gcd of two integers is the largest positive integer
that is a factor of both x and y
Equal to product of factor^min{multiplicity x, multiplicity y) for each factor in x or y
Least Common Multiple (LCM) - ANS-Smallest positive integer that is an integer
multiple of both x and y
Equal to product of Equal to product of factor^max{multiplicity x, multiplicity y) for each
factor in x or y
Relatively Prime - ANS-Two numbers are relatively prime if their greatest common
factor is 1
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