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Discrete Mathematics
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Gravity
Terms in this set (45)
Closure
a+b and a*b are integers
Commutative laws
a+b=b+a and a
b=b
a
Associative Laws
(a+b)+c=a+(b+c) and (a
b)
c=a
(b
c)
Distributive laws
(a+b)
c=a
c+b*c
Identity elements
a + 0 = a
a * 1 = a
Additive inverse
For every integer a, there is an integer solution x to the equation a+x=0
(This integer x, is called the additive inverse of A and is denoted by -a)
Cancellation law
If a
c=b
c with c not 0, then a=b
Ordering of integers
If b-a is a positive integer, then a<b
If a<b, then we can also write b>a
Positive integers
a is a positive integer iff a>0
Closure for positive integers
If a and b are positive, then a+b and a*b are positive
Trichotomy law
For every a, either a>0, a<0, or a=0
The well-ordering property
Every no empty set of positive integers has a least element
Summation notation
...
a divides b
If a and b are integers with a not equal to zero, we say that a divides b (a | b) If there is an integer c such that ac=b
If a divides b, we also say that a is a divisor or factor of b
The Division Algorithm
If a and b are integers such that b>o, then there are unique integers q and r such that a=bq+r with 0<=r<=b
Binary numbers
Base 2
Prime number
A positive integer greater than 1 that is divisible by no positive integers other than 1 and itself
Hexadecimal numbers
Base 16
Composite number
A positive integer that is greater than 1 that is not prime
Greatest Common Divisor
...
Mutually Relatively prime
a1, a2, a3,...,an
gcd(a1,...,an)=1
Relatively prime
...
Linear combination
...
Pairwise relatively prime
b1,b2,...,bn
gcd(b1,bn)=1
gcd(b2,b3)=1
gcd(bi,bj)=1
i not equal to j
Principle of Mathematical Induction
If for any statement involving a positive integer, n, the following are true:
1) The statement holds for n=1, and
2) Whenever the statement holds for n=k, it must also hold for n=k+1
Then, the statement holds for all positive integers, n.
Proof by Contradiction
...
Biconditional Proof
...
a is congruent to b modulo m
...
Congruence classes modulo m
...
Least nonnegative residue
...
Complete system of residues modulo m
...
Least nonnegative residues modulo m
...
Fermat's Little Theorem
If p is prime and a is a positive integer with p does not divide a, then a^p-1=1(mod p).
A set
A set is a specified unordered collection of elements
Pairwise disjoint family of sets
A family of sets is pairwise disjoint if for every 2 sets in the family, the intersection is an empty set
An intersection of sets
A set of the elements that two sets have in common
The principle of inclusion and exclusion
The principle that the cardinal its of sets may have too many terms so some may be included and some may be excluded
P-position
Position that is winning for the previous player
N-position
Position that is winning for the next lpayer
Planar graph
A graph that can be embedded in the plane
i.e. It can be drawn on the plane in such a way that it's edges intersect only at their endpoints; no edges cross each other
Isomorphic graphs
Two graphs which contain the same number of vertices connected in the same way.
K-colorable
A graph is k-colorable if it has a proper k-coloring
K-coloring
A labeling f:v(G)->S,|S|=K.
Degree-sum formula
The number of vertices times the degree of each vertex all divided by 2
x(K n,n)
Chromatic number of k on n vertices of a bipartite graph.
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