General Notes
:: Axiomatic Method
“in an axiomatic system, we never try to reproduce the facts in full, but only that side of them which is important or relevant in a particular context. This process of selecting what is relevant and disregarding everything else is the very essence of abstraction” (pinter page 23).
:: Euclid’s Elements
Definitions:
Def.1) Points exist
Def.2) A line exists between two points
Def.3) A plane exists between three or more points
Def.4) When two lines touch, angles are formed. When all angles are equal they are perpendicular.Def.5) With a center, A, and a radius line, AB, a circle can be created. All radii of the circle are equal.
Postulates:
(1) Between any two points and line can always be drawn.(2) Given a straight line you can always extend it by adding another point.(3) You can make a circle with any two points.(4) All right angles are always equal.
Axioms:
(Axio. 1) Things which are equal to the same thing are also equal to one another.If A = B AND B = C THEN A = C.
(Axio. 2) If equals be added to equals, the wholes are equal.If A = C AND B = D THEN A + B = C + D
(Axio. 3) If equals be substracted from equals, the remainders are equal.If A = C AND B = D THEN A – B = C – D
(Axio. 4) Things which coincide with one another are equal to one another.
:: Geometry
Circle Equation:
(x-h)^2 + (y-k)^2 = r^2
Where:h is location on x axisk is location on y axisr is radius
Potential Solutions:
An equation of the form x = a, where a is a number, has one solution.An equation of the form a = a, where a is a number, has infinite solutions.An equation of the form a = b, where a and b are diff numbers, has no solutions.
The way to think of this is that both sides of the equal sign are a plot on a graph. If there is a single point of intersection that is one solution. If there are no intersections, there are no solutions. If there the lines keep intersecting in a pattern there are unlimited solutions.
Second Derivative Test:
1: If f'(-2) = 02:If f”(-2) > 0, the graph of f has a minumum point x = -2If f”(-2) < 0, the graph of f has a maxiumum point x = -2If f” = 0, the test is inconclusive
Set Theory
Calculus
:: Differential Calculus
Limits:
lim delta x -> 0 dela x/delta y = dy/dx # Derivative notation
Formal:
lim f(x) = L # The limit of the function of x = Limit (L)x->c # When x approaches a specific point (c)
You can get f(x) closer to L by getting x closer to c. There will always be some gap, but it can basically be infintesamally small.
Epsilon: How close the range around f(x) is to L (episilon symbol) is proportionate to how close the range around x is to C (delta symbol).
One sided limits:
These one sided limits basically occur when a graph is broken. In these cases at the break point there is a difference in what point the graph is approaching, and so the limit can be different on either side of a plotted function.
lim f(x)x -> 2^- # This means the limit as x approaches 2 from the negative direction.
lim f(x)x -> 2^+ # This means the limit as x approaches 2 from the positive direction.
If the limit of a function grows sharply as x approaches a given point, it can sometimes be considered that it = infinity or that it does not exist and the function is unbounded.
When there is a difference between left-sided and right-sided limits, then the two sided limit is considered not to exist.
asymptote: ????
f(x) = |x-3|/x-3 # This will result in positive values if x is greater than 3, or negative values if x is lesser than 3.
Algebra
Algebra began by being refered to as the science of solving equations, but later became the math of sets and operations on sets.
Algebraic structure: arbitrary set whith one or more operations defined on it. Algebra is the study of these sets and operations.
Linear equation: ax + b = 0
Quadratic equation: ax^2 + bx + c = 0
Cubic Equation: x^3 + ax^2 + bx = c
Quartic equation: x^4 + ax^3 + bx^2 + cx = d
Tensor:???????
(a) # Zero dimeinsional array, aka scalar(a,b) # One dimensional array, aka a vector(a,b)(a’,b’) # Two dimensional array, aka a matrix (Note that matrices can have more than two dimensions.
:: Vectors
in physics vector needs magnitude and direction. Common notation
->v = (5,0) # This vector is moving 5 in the horizontal direction and 0 in the vertical. The notation is a lower case letter with an arrow above it.
-> (5) # This is the same vector as the one above.v= (0)
-> (5) # Diagonal vector, with its end point 5 along the x-axis and 3 along the y-axisv= (3)
R^2 with backbone = real coordinate space in two dimensions. The R means real coordinate space, and the power is dimensions. All possible 2-tuples. So basically the space that has all possible values in the vector (x,y) where x and y may be any number.
->v E R^3 # v is a member of the 3 dimensional coordinate space.
Adding Vectors:
??? More here ???
-> ->a + b =
Dot Product: (a,b) · (a’, b’) = aa’ + bb’ # Multiply corresponding components then add.
:: Matrix Algebra
[a,b] [c,d] · [a’,b’,c’][e,f] · [d’,e’,f’]
3*2 2*3
For two matrices to be multiplied the number of columns in matrix one needs to match the number of rows in matrix two. In the above case they are both 2, so it can be multiplied.
The end matrix size will be the value of the outside values (3*3), so the final matrix should be 3 x 3.
[(a*a’)+(b*d’), (a*b’)+(b*e’), (a*c’)+(b*f’)] # Row 1 of matrix 1 * each column of matrix 2[(c*a’)+(d*d’), (c*b’)+(d*e’), (c*c’)+(d*f’)] # Row 2 of matrix 1 * each column of matrix 2[(e*a’)+(f*d’), (e*b’)+(f*e’), (e*c’)+(f*f’)] # Row 3 of matrix 1 * each column of matrix 2
Commutative law of multplica-tion, AB = BA, is not true for matrices as the arrangement of columns and rows matters.
:: Boolean Algebra
A and B are subsets:
A + B for A u BA · B for A n B
:: Axioms of Algebra – Pinter page 21 – ??? Do More Work ???
Assumptions:A is any set* is an operation on A$ is a different operation on A (normally represented with _|_ symbol)
(1) a * b = b * a # Commutative, basically that the values of a and b are independent of order.(2) a * (b * c) = (a * b) *c(3) e * a = a and a * e = a for every a in A(4) a * a^-l = e and a^-1 * a = e(5) a * (b $ c) = (a * b) $ (a * c)
Pinter page 25
(Symbol: Circle with cross in the middle) = Tensor product
