BSc Mathematics SEM VI 2016 17 2016-17 Maths Paper III R Question Paper - Mumbai University | munotes

Questions asked in this paper

• (2) Figures to the right indicate marks

1. Q1 (a) Attempt any One from the following: 8 marks
   • (i) If in a metric space (X,d), for every decreasing sequence {F;,} of non-empty closed sets with —> 0, we have is a singleton set then prove that (X,d) is
   • (ii) If R are such that x < y then show that there exists a rational number r Q
   • (b) Attempt any Two from the following: 12
   • (i) Prove that the set of real numbers R is complete with respect to the usual distance
   • (ii) Define a complete metric space. If (X,d) complete metric space and Y is a closed subspace of X then prove that is complete
   • (iii) Check if Cantor’s Theorem is applicable in the following examples. Also , find in each case, where is a sequence of subsets of Rand the distance in R is usual (II) =
   • (iv) Show that the function f —+ R, defined by f(x) = (x — a)?(x — b)? + takes the value for some value of R. (distance in R being usual.)
2. Q2 (a) Attempt any One from the following: 8 marks
   • (i) Let (X,d) —> be a function. Prove that f is continuous on X if and only if for each open subset G of Y, is an open subset of X
   • (ii) Let (X,d) be a complete metric space. If T : X —> X is a contraction, then prove that T has a unique fixed point, 4 a unique point X such that T(x) =
   • (b) Attempt any Two from the following: 12
   • (i) (X,d) and be metric spaces. If f : (X,d) —> (Y,d’) is continuous on X then prove that for every A C X, C f(A). Also show that the inequality may be
   • (ii) Prove that every function f : (N,d) —> (Y,d’) where d is the usual metric and is any metric space is continuous
   • (iii) Discuss the uniform continuity of f : [1,00) —> R, defined by f(x) =
   • (iv) Let (0,00) be a continuous function, where (X,d) is a compact metric space. Show that 4 > Osuch that f(r) Va X
   • Q. P. Code: 05027
3. Q3 (a) Attempt any One from the following: 8 marks
   • (i) (X,d) is a metric space and A, B are subsets of X such that A is connected and B CA. Prove that B is connected. Give an example to show that if AC and A,C are connected then B need not be connected
   • (ii) Prove that a path connected subset of IR” (distance being Euclidean) is connected
   • (b) Attempt any Two from the following: 12
   • (i) If (X,d) be a connected metric space and X Z (distance in Z being usual distance) is a continuous function then prove that f is a constant function
   • (ii) Show that the set = < 2,1 < y <5} is a convex set in (R?, d) where d is the Euclidean distance
   • (iii) (X,d) is a connected metric space which is not bounded. Prove that for each ro X and for each r > 0 the set {2 X = r} is nonempty
   • (iv) Let A be the union of the following subsets S and L of R? Show that A is connected (distance in R* being Euclidean)
4. Q4 Attempt any Three from the following: 15 marks
   • (a) f : —> Ris a continuous such that f takes only rational values then show that f is
   • (b) Prove that (0,1) as of (IR,d) (d being usual distance) is not complete but is complete as.a subspace of where is discrete metric
   • (c) Let f : [a,b] — is continuous on [a,b] and differentiable on (a,b). If dc R with 0<c< that |f’(x)| <c, Vx (a,b) then prove that f is a contraction of [a,
   • (d) Let (X,d) and (Y, be metric spaces. Show that if f : X —> Y is uniformly continuous on X and if in X is Cauchy then show that the sequence is Cauchy in Y (ec) Show that E = — y? = 1} is path connected
   • (f) Prove or disprove : If A, B are connected subsets of R with respect to usual distance and then AN B is also connected

Read from the scan above, so a character or two may differ. The scan is the original.