MT1940 (Analysis) Definitions 4.0 / 5 based on 1 rating ? MathematicsAnalysisUniversityNone Created by: wxrpaintCreated on: 08-01-16 16:00 X subset of Y every element of X is an element of Y 1 of 35 intersection of X and Y every z that is an element of X and an element of Y 2 of 35 X union Y every z that is an element of X or an element of Y 3 of 35 X\Y (X minus Y) every z that is an element of X but not an element of Y 4 of 35 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 5 of 35 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 6 of 35 x is a maximum of S x is an upper bound of S and x is in S 7 of 35 P==Q (three lined equivalence sign) P is true exactly when Q is true 8 of 35 ¬P the negation, not P, is true when P is false 9 of 35 P ^ Q P and Q are both true 10 of 35 P v Q P is true, Q is true, or both are true 11 of 35 P -> Q P implies Q 12 of 35 absolute value |x| x if x greater than/equal to 0; -x if x less than 0 13 of 35 sequence a function x from N to R 14 of 35 increasing sequence (a sub n) for all n in N, a sub n+1 greater than or equal to a sub n 15 of 35 strictly increasing (a sub n) for all n in N, a sub n+1 greater than a sub n 16 of 35 monotonic sequence it is increasing or decreasing 17 of 35 (x sub n) bounded above some u in R such that x sub n less than/equal to u for all n in N 18 of 35 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 19 of 35 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. 20 of 35 (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 21 of 35 properly divergent lim(n to infinity)(x sub n) = (-)infinity 22 of 35 (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 23 of 35 m-tail of (x sub n) (x sub m+1, x sub m+2,...) 24 of 35 null sequence lim(n to infinity)(a sub n) = 0 25 of 35 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) 26 of 35 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 27 of 35 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 28 of 35 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 29 of 35 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 30 of 35 continuity (cluster point def.) if c is cluster point of A and limf(x) = f(c), or c is not cluster point 31 of 35 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 32 of 35 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) 33 of 35 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 34 of 35 absolute maximum some t in A such that f(t) greater than/equal to f(x) for all x in A 35 of 35
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