# 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

### Card 4

#### Front

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

#### Back

### Card 5

#### Front

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

#### Back

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