MT1940 (Analysis) Definitions

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  • Created by: wxrpaint
  • Created on: 08-01-16 16:00
X subset of Y
every element of X is an element of Y
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intersection of X and Y
every z that is an element of X and an element of Y
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X union Y
every z that is an element of X or an element of Y
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X\Y (X minus Y)
every z that is an element of X but not an element of Y
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S is bounded above (S subset of R)
there is some u in R such that s less than/equal to u for all s in S
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u is a supremum of S
u is an upper bound of S and u less than/equal to w for all upper bounds w of S
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x is a maximum of S
x is an upper bound of S and x is in S
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P==Q (three lined equivalence sign)
P is true exactly when Q is true
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¬P
the negation, not P, is true when P is false
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P ^ Q
P and Q are both true
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P v Q
P is true, Q is true, or both are true
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P -> Q
P implies Q
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absolute value |x|
x if x greater than/equal to 0; -x if x less than 0
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sequence
a function x from N to R
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increasing sequence (a sub n)
for all n in N, a sub n+1 greater than or equal to a sub n
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strictly increasing (a sub n)
for all n in N, a sub n+1 greater than a sub n
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monotonic sequence
it is increasing or decreasing
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(x sub n) bounded above
some u in R such that x sub n less than/equal to u for all n in N
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bounded sequence (a sub n)
there is some M in R with M greater than zero such that |a sub n| less than/equal to M for all n in N
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subsequence of (x sub n)
for n1 less than n2 less than n3... strictly increasing seq. of natural numbers, (x sub n sub i) is subseq.
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(x sub n) tends to infinity
for all a in R, there is some k in N such that, if n greater than/equal to k, x sub n greater than a
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properly divergent
lim(n to infinity)(x sub n) = (-)infinity
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(a sub n) converges to a
for all E greater than 0, there is some N in N such that, for all n greater than N, we have |a sub n - a| less than E
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m-tail of (x sub n)
(x sub m+1, x sub m+2,...)
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null sequence
lim(n to infinity)(a sub n) = 0
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S dense in R
for all x in R and for all E greater than 0, there is some s in S in the interval (x-E, x+E)
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periodic decimal (+-A.a1a2a3a4...)
there exists some k,m in N such that a sub n = a sub n+m for all n greater than/equal to k. m is the period
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cluster point c in R, A subset of R
every interval (c-d, c+d) contains a point x in A, with x not =c
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E-d def. of functional limits (lim f(x) = L as x to c; f: A->R)
for all E greater than 0, there exists d greater than 0 such that 0 less than |x-c| less than d -> |f(x)-L| less than E
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seq. def. of functional limits (lim f(x) = L as x to c; f: A->R)
for every sequence (x sub n) converging to c, where each x sub n in A and x sub n not =c, (f(x sub n)) converges to L
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continuity (cluster point def.)
if c is cluster point of A and limf(x) = f(c), or c is not cluster point
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continuity (E-d def.)
for all E greater than 0, there exists d greater than 0 such that for all x in A with |x-c| less than d, we have |f(x)-f(c)| less than E
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continuity (seq. def.)
for every sequence (x sub n) converging to c (x sub n in A for all n in N), (f(x sub n)) converges to f(c)
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bounded function f: A-> R
there exists M greater than 0 such that, for all x in A, |f(x)| less than/equal to M
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absolute maximum
some t in A such that f(t) greater than/equal to f(x) for all x in A
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Other cards in this set

Card 2

Front

every z that is an element of X and an element of Y

Back

intersection of X and Y

Card 3

Front

every z that is an element of X or an element of Y

Back

Preview of the back of card 3

Card 4

Front

every z that is an element of X but not an element of Y

Back

Preview of the back of card 4

Card 5

Front

there is some u in R such that s less than/equal to u for all s in S

Back

Preview of the back of card 5
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