Math Lecture notes
Definitions:
Relation – R subset of a product set ( A x B ) = can be seen as an area, different from function
where each x maps to one y , i.e. x<= y, or x=y^2
Function – is relation a->b where each x relates to one y. i.e. for y=sqrt(x) only take positives
- A is the domain
- B is the range
- b = f(a) for (a,b) subset Rf
- Image: f(a0)
- Inverse Image f-1(b0)
Function
- Is “onto” if f(A) = B, every A has a B
o Surjective, surjection
o Can map to the same B
- Is one-to-one if a1=/a2 -> f(a1) =/ f(a2)
o Injective, injection
o Mapped to different elements
o Every a has one b
o Can have b elements without an a
- Is bijection (one-to-one correspondence) if onto and one-to-one f: a<->b
o Is invertible
o f(f-1(b)) = b
o f-1(f(a)) = a
Bounds
- Consider set, A subset R where R is set of real numbers
- Supremum = least upper bound
o If all other bounds s’ have s<s’
- Infimum = greatest lower bound
- Least Upper Bound Property = R has property where every A bounded from above has
supremum in R,
o doesn’t hold in other sets, i.e. rational numbers Q {p/q where p is integer, and q
is natural}
o example: A { x rational : x^2 < 2 } , does not have least upper bound, since the
bound s = sqrt(2) is not rational

but is fine if is set of real numbers
Crucial concept of Convergence
- sequence does not move much for large N
- set is big enough that there is an element to converge to
- Then have Sequence R^n = R x R x R x R
- Sequences of functions
Metric Spaces
- Topological Spaces – distances are well defined
- Metric Space = set X associate with distance function d : X x X -> R+ such that
o D(p,q) = 0  p=q
o Symmetry d(p,q) = d(q,p)
o Triangle inequality: d(p,q) <= d(p,r) + d(r,q)
 Shortest path is if r=0
- Examples:
o R real line: metric d(x,y) = | x – y |
o Euclidean distance: R^2 with (d(x1,y1),(x2,y2)) = sqrt( (x1-x2)^2 + (y1-y2)^2 )
o Set of rationals with d(x,y) = 0 if x=y and 1 otherwise
Metric Spaces: other concepts
- Metric Space X, point x, set E
- Open and closed sets,
o open where there is r st y : d(x,y) < r
o like a big circle around the set
- x limit point
- x boundary point
- E is open set if no boundary points
- E is closed if contains all boundary points
- E is dense in X
o If every point in E can be approximated by an x, less than epsilon e
- Set is bounded if
o d(x,q) <= M
Metric Spaces
- Sequence: f : N -> X
- Sequence converges to point x, if
-
-
o D(xn, x) < epsilon
o Write lim xn where n-> inf = x or lim xn = x
Cauchy sequence
o For each eps > 0, exists N : M >= N => d(xN, xM) < esp
Metric space is “complete” if all Cauchy sequences converge within X
o R real line is complete
o Q rationals is not complete
Cauchy -> sequence where gets arbitrarily closer