2 N A Sx B. 6 a2 −b2 =(ab)(a−b) 7 a3 −b3 =(a−b)(a2 ab b2) 8 a3 b3 =(ab)(a2 −ab b2) 9 a n−bn=(a−b)(an−1 a −2b an−3b2 bn−1) 10 an= aaantimes 11 ama n= am 12 am an = am−nif m>n =1 ifm=n = 1 an−m if m. 2 SA, because A\ ;.
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Let n(A −B) = 25 x, n(B −A) = 2x and n(A ∩B) = 2x , If n(A) =2(n(B)), then ' x ' is Let n ( A − B. 8 x 2 O 9 r > 0 st x 2 B( x;r ) O (ii) Any set U M containing a ball B( x;r ) about x is called neighborhood of x The collection of all neighborhoods of a given point x is denoted by U (x ) Remark 3 The collection M= fO M jO is open g is a topology on M Theorem 4 (Induced/Relative Metric). NOTES ON METRIC SPACES JUAN PABLO XANDRI 1 Introduction Let X be an arbitrary set, which could consist of vectors in Rn, functions, sequences, matrices, etc We want to endow this set with a metric;.
A 2 x 2 b 2 a 2 xb 2 = a 2 xb 2 • (x 1) Equation at the end of step 2 a 2 xb 2 • (x 1) = 0 Step 3 Theory Roots of a product 31 A product of several terms equals zero When a product of two or more terms equals zero, then at least one of the terms must be zero.
Restriction of a convex function to a line f Rn → R is convex if and only if the function g R → R, g(t) = f(xtv), domg = {t xtv ∈ domf} is convex (in t) for any x ∈ domf, v ∈ Rn can check convexity of f by checking convexity of functions of one variable. You want $x = a^2 b^2 = (ab)(ab)$ Let $m = ab$ and $n = ab$, then note that $a = (mn)/2$ and $b = (nm)/2$ For these to be natural numbers, you want both $m$ and $n$ to be of the same parity (ie, both odd or both even), and $m \le n$ For any factorization $x = mn$ satisfying these properties, $a = (mn)/2$ and $b = (nm)/2$ will be a solution. N(BA)=3x and n(A B)=x Draw a diagram to illustrate this information If n(A)=n(B) Find (i) the value 3. Ie a way to measure distances between elements of XA distanceor metric is a function d X×X →R such that if we take two elements x,y∈Xthe number d(x,y) gives us the distance.
