Econ2125/6012, Semester-1 2023 Assignment-1

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Assignment-1

Econ2125/6012, Semester-1 2023

Each question is 5 marks.

Question-1: For the Euclidean space of RK define a new metric as follows:

d (x, y) = x(K)| xi − yi |p

Show that the Euclidean space of RK with this metric is a metric space.

Question-2: For x cR1 and y cR1 , define

d1 (x, y ) =  (x − y )2

d2 (x, y ) =

d3 (x, y ) = |x2   − y |2

d4 (x, y ) = |x − 2y |

d5 (x, y ) =

Determine. for each of these, whether d is a metric or not.

Question-3: True or False? Justify your answer.

(i) Union of a convex and a non-convex set can be convex.

(ii) Every closed and bounded set in a metric space is compact.

(iii) If S is an open set andf is a continuous function over S,f(S) is open as well.

(iv) Any finite set of numbers in R is closed.

(v) Any closed subset of a metric space is a complete space.

Question-4: For each of the following sets, decide whether or not the set is a) closed, b) open, c) compact, d) connected. Justify your answer.

(i) [5,6) U (6,7]

(ii) {(x, y, z) = R3  | x2 + y2 + z2  > 9}

(iii) {(x, y) =Q | x2 + y2 共 1}

(iv) The following sets in R2

(v)

Question-5: Prove that set S is not connected if

(i)         it can be written as union of two closed disjoint set.

(ii)        It can be written as union of two open disjoint sets.

Question-6: Remember that a set S in a metric space is a closed set iff for every sequence {xn } that converges to x, x is in S. Using this definition show that why interval (0,1) is not a closed set.

Question 7: Decide which of the following sets is convex or non-convex. Justify your answer.

(i){(x, y) R2  | x + y < 2}

(ii){(x, y) R2  | 1 < x2 + y2  < 4}

(iii){x Q | 0 < x < 1}

(iv) B - A if B and A are convex sets in RK

(v) The following shape in R2

Question 8: Let M be a metric space. If {xn } and {yn } are sequences in M such that xn → x and yn → y, show that d (xn , yn ) → d (x, y ).

Question-9:  Suppose  A  utility  function U(x) = {x1(p) + (1−)x2(p)}1/ p with 0<< 1 and p>0 is defined over the set S = {(x1 , x2 ) | x1p1 + x2p2 M} where p1 , p2   are prices. Prove that for every positive p1 , p2  and M, there exists a utility maximizing level of consumption of (x1 , x2 ) .

x3

Question-10: Consider the function f : R → R defined by f (x) =         2  , using the epsilon-

1 + x

delta definition of a continuous function, prove that this function is continuous on R.