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Glossary of Math Econ

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Created by mjmelanc

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Natural Numbers
The Counting Numbers, not including zero
Integer
Natural Numbers, their negatives, and zero
Rational Numbers
expressible as a fraction where the numerator and denominator are integers (except 0 as the denominator)
Irrational numbers
cannot be expressed as fractions
Real Numbers
All rational plus irrational numbers
Set
collection of objects, S = {1,2,3}, or x is an element of N such that x < 4
Strong Subset
A set within a set which may not incorporate the entire set
Intersection
Element “x” is an element of both B and C
Union
Element “x” is an element of either B or C
Complements
B A = x is an element of B but not A
(Euclidean) Real Line
A set of all ordered n-tuples of Real Numbers
Linear Independence
Let {c1, c2, …, cn} be elements of R and let {v1, v2, …,vn} be elements of R^n, which is linearly independent iff all elements of R = 0, when c1v1+c2v2….+cnvn = 0.
Row Rank
largest number of L.I. row vectors
Column Rank
Largest number of L.I. column vectors
Rank(A)=
Column Rank(A) = Row Rank(A)
Rank A
min
Full Rank
m=n=rank(A); rank(A) = min{m,n}
If Rank(A) equals number of equations
Infinite Solutions
If Rank(A) equals number of solutions
Either 0 or 1 Solution
If Rank(A) equals equations and solutions
Exactly 1 Solution
Matrix Properties
A(BC) = (AB)C; A(B+C) = AB + AC; AB=/=BC
(AB)’=
B’A’
Matrix Inversion
2x2 matrix: 1/(det(A)*(n2,2 –n1,2; -n2,1, n1,1)
(A^-1) ^-1 =
A
(A^-1)^T =
(A^T)^-1
(AB)^-1 =
B^-1 A^-1
Convergent

Tend to Finite Value
Divergent
Tend to increase without bound
Oscillating
alternate increasing and decreasing
Formal Distance
e>0, Ie(r) = {s in set R such that abs(s-r) < e}
There is a limit if
for any e>0, there exists an Ne >0 such that for all n >= Ne, xn exists in set Ie(r)
Distance Metric =
||x-y|| = sqrt((x1-y1)^2+(x2-y2)^2+….(xn-yn)^2)
Epsilon ball around the limit =
x in set R^m such that the distance between x and r < e
A set S is open if
for all x in set S there exists an e > 0 such that Be(x) is a strong subset of S
A set is closed iff
Its compliment is open.
A set is compact if
it is closed and bounded
A function is continuous if
a fn that maps x to y is cts at x0 iff for any e>0 there exists a de >0 such that x is an element of Bde with respect to x0 (Bde(x0)), which implies that f(x) is an element of the Bde wrt x0.
Functions are linear iff
f(ax+by) = a(f(x))+b(f(y)) for all scalars (a,b) and all (x,y) are elements of All Real Numbers in the Euclidean Space of the length 2*m
A set is bounded iff
Any x1 is an element of the set, any x2 is an element of the set such that the distance from x1 to x2 equals the ||x1-x2|| < B which is an element of R.

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