analysis definitions

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analysis definitions
2013-05-17 10:42:25

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  1. Fundamental theorem of arithmetic
    Every positive integer except 1 can be expressed uniquely as a product of primes
  2. Bernoulli's inequality
    (for all n in N)(For all a >1)[(1+a)^n>=1+na]
  3. A set S in R is said to be bounded above if
    (There exists k in R)(For all x in S)(x<=k)
  4. Supremum
    • Definition of upper bound and least upper bound.
    • ie (There exists e>0)(There exists b in S)(b>a-e)

    We write a=sup S
  5. infimum
    • a is a lower bound for S; that is
    • (for all x in S)(x>=a)
    • a is the greatest lower bound for S; that is
    • (for all e >0)(There exists b in S)(b<a+e)
    • b=inf S
  6. Let R be a relation from X to Y. The domain of R is the set
    D(R)={x in X |there exists y in Y | (x,y) in R}
  7. The range of R is the set
    Ran(R)={y in Y |there exists x in X | (x,y) in R}
  8. The inverse of R is the Relation R^-1 from Y to X
    R^-1= {(y,x) in YxX | (x,y) in R}
  9. A relation R on X is said to be reflexive if
    (For all x in X) (x,x) in R
  10. R is said to be symmetric if
    (For all x in X)(For all y in Y){[(x,y) in R] -> [(y,x) in R]}
  11. R is said to be transitive if
    (For all x,y,z in X){[((x,y) in R) and ((y,z) in R)] -> [(x,z) in R]}
  12. Equivalence relation
    If it is symmetric, transitive and reflexive
  13. Let R be an equivalence relation on X. Let x be in X. The equivalence class of x wrt R is the set
    [x]R = {y in X | (y,x) in R}
  14. F:X->Y and A in X. Image of A under f is the set
    f(A) = {f(x) | x in A}
  15. f:X->Y. Then f is called injective
    (For all x1 in X)(For all x2 in X)[(x1 not equal to x2) -> (f(x1) not equal to f(x2))]
  16. f: X->Y, then f is called a surjection if
    • (for all y in Y)(there exists x in X)[f(x) =y]
    • This means Ran(f)=Y
  17. f:X->Y is called a bijection if
    It is both an injection and a surjection
  18. The Archimedian Principle
    Let x be in R. Then there exists n in Z such that x<n
  19. A sequence converges to a limit a in R if
    (For all e >0)(There exists N in N+)(for all n in N+)[(n>N)->(|an-a| <e)
  20. The sequence an diverges to infinity if
    (For all M in R)(There exists N in N+)(For all n in N+)[(n>N)->(an<M)]
  21. If a sequence of real numbers is bounded above and increasing then it is
  22. If a sequence of real numbers is bounded below and decreasing then it is
  23. If the series Sum (an) from n=1 to infinity is convergent then
    lim n-> infinity is 0