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Definition
a relation satisfying two properties:
1. for all x,y x>y,y>x or y=x
2. if x |
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| a set with an order defined |
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| bounded above (and upper bound) |
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Definition
an ordered set S with E as a subset of S:
if there exists a "beta" such that "beta">=x for all x in E |
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| "alpha" is the supremum of E if it is an upper bound, and it's <= all the upper bounds |
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| least-upper-bound property |
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Definition
an ordered set S
1. E is a subset of S
2. E is non-empty
3. E is bounded above
then sup(E) exists in S |
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| field is a set F with two operations (addition and subtraction) that satisfy the field axioms (name them, there are 11) |
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| a field that is also an ordered set |
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there exists an ordered field R which has the least-upper-bound property
Moreover, R contains Q as a subfield.
Subfield means that Q is a subset of R, and the operations are preserved |
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| Schwarz Inequality (in words) |
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| each pair of corresponding components multiplied, then added together |
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| the norm of a vector is the square root of a vector "dotted" with itself |
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| for every x in A there is an associated f(x) in B |
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| domain, range, codomain, preimage |
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| (do it yourself, tis is easy) |
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| if x1 does not equal x2, then f(x1) does not equal f(x2) |
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| if the range of the function equals the codomain |
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| if there exists a 1-1, onto mapping from A to B |
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| finite, infinite, countable, uncountable, at most countable |
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| a function f defined on the set of all natural numbers |
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| the set with all x such that x is in at least one of the things being unioned together |
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| the set of x's such that x is in all of the things being "intersectioned" together |
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| A and B intersect (are disjoint) |
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| if A intersect b is non-empty (empty) |
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Definition
a set (with elements we call points) and an associated "metric" or "distance" that obeys:
1. distance between two points is greater than - if the points aren't equal, and zero if they are
2. doesn't matter which order you take the pioints in
3. d(p,g)<=d(p,r)+d(r,q) |
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a set E subset of R^k is convex if
"lambda"x + (1-"lambda")y is an element of E
whenever x,y are in E and lambda is between 0 and 1 |
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