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Math
Discrete Math
Quiz 2 Discrete
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Terms in this set (46)
N
set of positive integeres
prime
An integer n is prime if, and only if, n > 1 and for all positive integers r and s, if n=rs, then either r or s equals n.
∀positive integer sr and s,if n=rs
then either r = 1 and s = n or r = n and s = 1.
composite
An integer n is composite if, and only if, n>1 and n=rs for some integers r and s with 1<r<n and 1<s<n.
∃ positive integers r and s such that n = r s
and 1 < r < n and 1 < s < n.
counterexample
To disprove a statement of the form "∀x ∈ D, if P(x) then Q(x)," find a value of x in D for which the hypothesis P(x) is true and the conclusion Q(x) is false.
method of exhaustion
proving every single one
method of generalizing from a generic particular
To show that every element of a set satisfies a certain property, suppose x is a particular but arbitrarily chosen element of the set, and show that x satisfies the property.
divisibility
The notation d|n is read "d divides n." Symbolically, if n and d are integers and d ̸= 0:
d |n ⇔ ∃an integer k such that n = dk.
The Quotient-Remainder Theorem
Given any integer n and positive integer d, there exist unique integers q and r such that
n=dq+r and 0≤r<d
proposition
for all a,b ∈ℤ if a·b=1, then a=b=1 or a=b=-1
floor and ceiling
x∈ℝ unique n∈ℤ s.t. n≤x<n+1
x∈ℝ unique n∈ℤ s.t.
n-1<x≤n
sequence
a function where f:D≤ℤ→ℝ
where D ={K ∈ℝ| m≤k≤n}
or all integers are greater than some n
is a function whose domain is either all the integers between two given
integers or all the integers greater than or equal to a given integer.
properties of the sums
you can do the sequence first and then multiply or vice versa
you can add the sums individually or together it will be the same
n!
k=1 pi n = 1·2·3·n
Principle of mathematical induction
asserts that P(k) being true implies P(k+1) is true.
prime number
The only divisors are one and itself
recurrence relation
defines each later term in the sequence by reference to earlier terms and also one or more initial values for the sequence.
is a formula that relates each term ak to certain of its predecessors ak−1, ak−2, . . . , ak−i where i is an integer with k − i ≥ 0.
Product Notation
m<n ak you multiply instead of divide.
a=b(modn)
n/a-b (definition)
well ordered properties
s in z. s not equal to zero. all elements of s are longer than a certain z in integer
principle of strong mathematical induction
Prove Basis step P(A), P(A+1), P(a+2)...P(b) where b≥a and b∈ℤ
IH assume p(k) ∀ k∈ℤ a≤k≤n and n∈ℤ
induction step
Prove P(N+1) and then by SPMI P(N) is true ∀n≥a
Prime number
a number > 0 and is only divisible by one and itself
Every number > 1 is divisible by a prime
Principle of mathematical induction
let a and n be integers.
1. assume P(A) is true
2. for all K≥a, if P(k) is true then p(k+1) is true
then statement is true for all integers n≥a, P(n) is true
well ordering principle of integers
1) every non empty sets of integers greater than some fixed integer a has at least one element.
2) must contain at least one elemenet
recurrence relation
is a formula that relates each term ak to certain of its predecessors where i is an integer with k − i ≥ 0.
Fibinooccia sequence
F1=1 F2 = 2 Fn = Fn-1 +fn-2 where n≥3
method of proof by division into cases
prove a statement in the form p1, p2, p3
set
a collection of elements
subset
A is a subset of B only if every element of A is in B
proper subset
If there is at least one element in B not in a
Universe
The largest set where the elements of our set will be taken from
Union
elements that are in A or B
there exists
intersection
elements that are in A and B
for all
difference
elements that are in A but not in B
Complement
Elements that are not in A
disjoint
sets have no elements in common A∩B=∅
mutually disjoint
{ai......aj} if A∩B = ∅ and if i≠j
Partition
A finite or infinite collection of nonempty sets {A1, A2, A3 ...} is a partition of a set A if, and only if,
1. A is the union of all the Ai
2. The sets A1, A2, A3, . . . are mutually disjoint.
power set
is the set of all subsets of A
cartesian product
A × B = {(a, b) | a ∈ A and b ∈ B} .
de morgans law
The negation of an and statement is logically equivalent to the or statement in which each component is negated.
The negation of an or statement is logically equivalent to the and statement in which each component is negated.
inverse relation
Let R be a relation from A to B. Define the inverse relation R−1 from B to A as follows:
R−1 ={(y,x)∈B×A|(x,y)∈R}.
relation on a set A
a relation from A to A.
reflexive
for all x ∈ A,x R x.
symmetric
for all x,y∈A,if xRy then yRx.
transitive
for all x,y,z∈A,if xRy and yRz then xRz.
R is an equivalent relation
if and only if R is reflexive, symmetric, and transivitive
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