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Fundamental theorem of arithmetic
Every positive integer except 1 can be expressed uniquely as a product of primes
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Bernoulli's inequality
(for all n in N)(For all a >1)[(1+a)^n>=1+na]
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A set S in R is said to be bounded above if
(There exists k in R)(For all x in S)(x<=k)
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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
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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
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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}
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The range of R is the set
Ran(R)={y in Y |there exists x in X | (x,y) in R}
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The inverse of R is the Relation R^-1 from Y to X
R^-1= {(y,x) in YxX | (x,y) in R}
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A relation R on X is said to be reflexive if
(For all x in X) (x,x) in R
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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]}
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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]}
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Equivalence relation
If it is symmetric, transitive and reflexive
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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}
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F:X->Y and A in X. Image of A under f is the set
f(A) = {f(x) | x in A}
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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))]
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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
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f:X->Y is called a bijection if
It is both an injection and a surjection
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The Archimedian Principle
Let x be in R. Then there exists n in Z such that x<n
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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)
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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)]
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If a sequence of real numbers is bounded above and increasing then it is
convergent
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If a sequence of real numbers is bounded below and decreasing then it is
convergent
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If the series Sum (an) from n=1 to infinity is convergent then
lim n-> infinity is 0
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