Show that Sn is path-connected, by constructing for any two points x, y € Sn an explicit path connecting them.
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- Does either of P = (4, 11, 20) or Q = (-1, 6, 16) lie on the path r(t) = ( 1 + t, 2 + t 2, t 4)?How many shortest lattice paths start at (2,2) and end at (15,15)?[ Preview end at (15,15) and pass through (10,7)? Preview end at (15,15) and avoid (10,7)? PreviewOf all lattice paths from (0, 0) to (9, 5) that only move up and to the right, how many: avoid the point (2, 3)? avoid the path between (1, 2) and (2, 2)? It could include either of those points - just not both.
- Suppose that a > 0 is a positive integer, and n > a. How many lattice paths are there from (0, 0) to (n, n) that do not go above the line y = x + a? hint: Catalan NumberFind image of a and preimage of b T(v1, v2, v3) = (4v2 – v1, 4v1 + 5v2) v = (2, -3, -1) , w = (3, 9)2. Suppose n ≥ 1 is an integer. Consider an (n + 1) x (n + 1) grid of integer points; i.e. points of the form (a, b) where 0 ≤ a,b ≤n. A Binomial Path with 2n steps is a path from the point (0, 0) to (n, n) formed by moving either 'right' (i.e. from (a, b) to (a +1, b)) or ‘up' (i.e. from (a, b) to (a, b+1)). (a) Draw all distinct Binomial Paths with 2n steps when n = = 2. (b) Write down a correspondence that relates the Binomial Paths with 2n steps to strings of length 2n consisting of exactly n 1s and n Os. More precisely, let B₁, be the set of Binomial Paths with 2n steps, and let Sn be the set of strings of length 2n consisting of exactly n 1s and n Os. Construct a bijection f: Bn Sn. (You don't have to prove that it is a bijection.)
- Prove (Menger) if x, y are vertices of a graph G and xy e E(G), then the minimum size of an x,y-cut equals the maximum number of pairwise internally disjoint x,y-pathsProve A strong path P from x to y is a strongest x - y path in the following cases. (i) P contains only a-strong edges. (ii) P is the unique strong x - y path.-. Let A= {(X₁4) & TR²² | xy ≤1₂ x+y >1}. -) al Graph A. Is A bounded? Is A path connected? b) Using equations and inequalities calculate c (A1, int(A), and bd (A).