## Glossary of Math Econ

Created by mjmelanc

- 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.