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College Mathematics

TIFR 2014 Problem 27 Solution -Homeomorphism of product space with mother space


TIFR 2014 Problem 27 Solution is a part of TIFR entrance preparation series. The Tata Institute of Fundamental Research is India’s premier institution for advanced research in Mathematics. The Institute runs a graduate programme leading to the award of Ph.D., Integrated M.Sc.-Ph.D. as well as M.Sc. degree in certain subjects.
The image is a front cover of a book named Topology by James.R. Munkres. This book is very useful for the preparation of TIFR Entrance.

Also Visit: College Mathematics Program of Cheenta


Problem:


(X) is a topological space with infinite cardinality which is homeomorphic to (X \times X). Then

A. (X) is not connected.

B. (X) is not compact.

C. (X) is not homeomorphic to a subset of (\mathbb{R})

D. none of the above


Discussion:


Let us take (X=\mathbb{Z}). The topology is the subspace topology obtained from (\mathbb{R}). Then, in fact, (X) has the discrete topology. This is because for every (n\in \mathbb{Z}), ((n-1,n+1)) is an open subset of (\mathbb{R}) and hence ((n-1,n+1)\cap \mathbb{Z}={n} ) is open in (\mathbb{Z}).

We have a bijection from (\mathbb{Z} \times \mathbb{Z}) to (\mathbb{Z}). The product of two discrete spaces is again a discrete space because we have all sets of the form ({n} \times {m} ={(n,m)} ) as open sets. Now, any map from a discrete space is continuous. Because the inverse image of any open set is a subset of the space and every subset is open in that space. So in fact, the bijection that we know of from (\mathbb{Z} \times \mathbb{Z}) to (\mathbb{Z}) is continuous in both ways. Therefore in this setup (X \times X) is homeomorphic to (X). And also (X=\mathbb{Z}) is a subset of (\mathbb{R}). So this provides us with a counterexample for option C.

Next, we will take (X=\mathbb{Z}) to disprove A and B together. But now we will consider a different topology for (\mathbb{Z}). We will consider the indiscrete topology. That is, the only sets which are open in (X) are (X) and the empty set. Note firstly, that since we only have finitely many open sets to start with, (X) is compact. (X) is also connected because the only non-empty set which is both open and closed is (X) itself. We use the same bijection that we know of between (\mathbb{Z}) and (\mathbb{Z}\times \mathbb{Z}). Note that if a map has its co-domain set as an indiscrete space then that map must be continuous. This is because the only open sets are empty and whole set, and their inverse images are empty and the whole domain set respectively which are always open. Also, note that if (X) has the indiscrete topology then so does (X \times X). Because the open sets in (X \times X) are (\phi \times \phi = \phi , \phi \times X = \phi , X \times \phi = \phi, X \times X ). So both the spaces in consideration have indiscrete topology. So the bijection that we have is continuous in both ways. This, therefore, gives the counterexample for A and B.


Helpdesk

  • What is this topic: Topology
  • What are some of the associated concept: Homeomorphism, indiscrete topology
  • Book Suggestions: Topology by James.R. Munkres
Categories
College Mathematics

TIFR 2014 Problem 26 Solution -Number of irreducible Polynomials

TIFR 2014 Problem 26 Solution is a part of TIFR entrance preparation series. The Tata Institute of Fundamental Research is India’s premier institution for advanced research in Mathematics. The Institute runs a graduate programme leading to the award of Ph.D., Integrated M.Sc.-Ph.D. as well as M.Sc. degree in certain subjects.
The image is a front cover of a book named Linear Algebra Done Right by Sheldon Axler. This book is very useful for the preparation of TIFR Entrance.

Also Visit: College Mathematics Program of Cheenta


Problem

The number of irreducible polynomials of the form (x^2+ax+b) , with (a,b) in the field (\mathbb{F}_7) of 7 elements is:

A. 7

B. 21

C. 35

D. 49


Discussion:


First, what is the number of polynomials of the form (x^2+ax+b) in (\mathbb{F}_7) ? (a) has 7 choices and (b) has 7 choices. So we have a total of (7 \times 7 =49) monic degree 2 polynomials in (\mathbb{F}_7).

How many of these are reducible? If a monic polynomial of degree 2 is reducible then it must break into two factors in the form (p(x)=(x-\alpha)(x-\beta) ).

We want to count the number of polynomials of the form (p(x)=(x-\alpha)(x-\beta)). This will give us the number of reducible polynomials and we will subtract it from the total number to get the number of irreducible.

For the counting purposes, we must be careful not to have repetitions. For (\alpha=0) we can allow (\beta=0,1,..,6). For (\alpha=1) we can allow (\beta=1,…,6) (NOT 0 because that will again give ((x-0)(x-0)) which we have already counted in the (\alpha=0) situation ).

For (\alpha=2) we have (\beta=2,…,6) and so on… for (\alpha=6) we have (\beta=6) only.

So in total, we have (7+6+…+1=\frac{7\times 8}{2} = 28 ) polynomials which are reducible, has degree 2 and is monic.

Therefore, the number of irreducible polynomials are (49-28=21).


HELPDESK

  • What is this topic: Linear Algebra
  • What are some of the associated concept: Monic polynomial
  • Book Suggestions: Linear Algebra done Right by Sheldon Axler
Categories
College Mathematics

TIFR 2014 Problem 24 Solution – Order of Generated Subgroup


TIFR 2014 Problem 24 Solution is a part of TIFR entrance preparation series. The Tata Institute of Fundamental Research is India’s premier institution for advanced research in Mathematics. The Institute runs a graduate programme leading to the award of Ph.D., Integrated M.Sc.-Ph.D. as well as M.Sc. degree in certain subjects.
The image is a front cover of a book named Contemporary Abstract Algebra by Joseph A. Gallian. This book is very useful for the preparation of TIFR Entrance.

Also Visit: College Mathematics Program


Question: 

Let (H_1, H_2) be two distinct subgroups of a finite group G, each of order 2. Let (H) be the smallest subgroup containing (H_1) and (H_2). Then order of (H) is

A. always 2

B. always 4

C. always 8

D. none of the above


Discussion:


Let’s check out the simplest example where we suspect things might get wrong.

Take (S_3) as the group. It has 3 order 2 subgroups. If we take any two of those then the subgroup generated by (i.e the smallest subgroup containing) should have order greater than 2; so it must have order 3 or 6 and that itself leads us to conclude that the answer is none of the above.

For example take (<(1 2)>=H_1,<(13)>=H_2). Then (H) contains ( (1 2), (1 3), (1 2)(1 3)=(1 3 2), (1 3)(1 2)=(1 2 3)). Therefore (H) has atleast 4 elements, so it must have 6 elements i.e, (H=S_3).


Helpdesk

  • What is this topic:Abstract Algebra
  • What are some of the associated concept: Order of a Subgroup
  • Book Suggestions: Contemporary Abstract Algebra by Joseph A. Gallian
Categories
College Mathematics

TIFR 2014 Problem 25 Solution – Identifying Isomorphic Groups


TIFR 2014 Problem 25 Solution is a part of TIFR entrance preparation series. The Tata Institute of Fundamental Research is India’s premier institution for advanced research in Mathematics. The Institute runs a graduate programme leading to the award of Ph.D., Integrated M.Sc.-Ph.D. as well as M.Sc. degree in certain subjects.
The image is a front cover of a book named Topics in Algebra by I.N.Herstein. This book is very useful for the preparation of TIFR Entrance.

Also Visit: College Mathematics Program


PROBLEM:TRUE/FALSE?


Which of the following groups are isomorphic?

A. (\mathbb{R}) and (\mathbb{C})

B. (\mathbb{R}^* ) and   (\mathbb{C}^*)

C. (S_3 \times \mathbb{Z}_4) and (S_4)

D. (\mathbb{Z}_2 \times \mathbb{Z}_2 ) and ( \mathbb{Z}_4 )


DISCUSSION:


(\mathbb{R}) and (\mathbb{C}) are isomorphic:

Both these spaces are vector spaces ove (\mathbb{Q}).

We state a result without proof.

Theorem: Let (V) be an infinite dimensional vector space over a countable field (F). Then the dimension of (V) over (F) is (|V|) (the cardinality of (V) ).

This immediately tells us that (\mathbb{R}) has dimension (|R| = 2^{\aleph _0 } ).

Now the dimension of (\mathbb{C}) over (\mathbb{Q}) is (dim(\mathbb{R}^2) = 2 \times 2^{\aleph _0 } =2^{\aleph _0 } ).

What did we just show? We showed that (dim(\mathbb{C})=dim(\mathbb{R}) ) over (\mathbb{Q}).

Therefore they are isomorphic as vector spaces. So they are isomorphic as groups with respect to addition.

(\mathbb{R}^* ) and   (\mathbb{C}^*) are not isomorphic:

(\mathbb{C}^) has an element of order 4, namely (i) has order 4. If there was an isomorphism then the corresponding element in (\mathbb{R}^) will also have order 4. But there is no element of order 4 in (\mathbb{R}^*). Hence we conclude that these two groups are not isomorphic.

(S_3 \times \mathbb{Z}_4) and (S_4) are not isomorphic:

The order of (((1 2 3), [1] ) \in S_3 \times \mathbb{Z}_4) is 12. (S_4) does not contain an element of order 12.

 (\mathbb{Z}_2 \times \mathbb{Z}_2 ) and ( \mathbb{Z}_4 ) are not isomorphic:

(\mathbb{Z}_2 \times \mathbb{Z}_2 ) is not cyclic and ( \mathbb{Z}_4 ) is cyclic. So they can not be isomorphic.


HELPDESK

  • What is this topic:Modern Algebra
  • What are some of the associated concept: Isomorphism
  • Book Suggestions: Topics in Algebra by I.N.Herstein
Categories
College Mathematics

TIFR 2014 Problem 22 Solution -An application of Intermediate Value Theorem


TIFR 2015 Problem 7 Solution is a part of TIFR entrance preparation series. The Tata Institute of Fundamental Research is India’s premier institution for advanced research in Mathematics. The Institute runs a graduate programme leading to the award of Ph.D., Integrated M.Sc.-Ph.D. as well as M.Sc. degree in certain subjects.
The image is a front cover of a book named Introduction to Real Analysis by R.G. Bartle, D.R. Sherbert. This book is very useful for the preparation of TIFR Entrance.

Also Visit: College Mathematics Program of Cheenta


Problem:


Let (f:\mathbb{R}^2 \to \mathbb{R} ) be a continuous map such that (f(x)=0 ) for only finitely many values of (x). Which of the following is true?

A. either (f(x) \le 0 )for all (x) or (f(x) \ge 0 ) for all (x).

B. the map (f) is onto

C. the map (f) is one-to-one

D. none of the above


Discussion:


Let (f) be the map (f(x_1,x_2)=x_1^2 +x_2^2 ). Then (f) is zero at only (0,0). (f) is continuous because (x=(x_1,x_2) \to x_1 \to x_1^2 ) is continuous from (\mathbb{R}^2 \to \mathbb{R} \to \mathbb{R} ). i.e, each arrow is continuous. The first arrow is the projection map, and such maps are always continuous, and the second arrow is just squaring, which is continuous. And composition of continuous functions are continuous, so (x \to x_1^2 ) is continuous function from (\mathbb{R}^2 \to \mathbb{R}). Where here and henceforth (x=(x_1,x_2)\in \mathbb{R}^2 ).

Similar reasoning will show that (x \to x_2^2 ) is continuous function from (\mathbb{R}^2 \to \mathbb{R}).

Sum of continuous functions is continuous, so the map (x \to x_1^2+ x_2^2 ) is continuous function from (\mathbb{R}^2 \to \mathbb{R}).

This function (f) is not one-one since (f(1,0)=f(0,1)=1) and it is not onto since it only takes values in ([0,\infty ) ).

So we now are sure that B,C are false options.

We will prove A now.

Let (x=(x_1,x_2)) and (y=(y_1,y_2)) be two points in (\mathbb{R}^2) such that (f(x)>0) and (f(y)<0).

We will prove that this will imply infinitely many zeros in between (x) and (y). But wait a second… what does between mean in this context? For that we consider the paths between (x) and (y). Note that there are infinitely many paths between any two points in (\mathbb{R}^2). Further, we can in fact have infinitely many paths completely disjoint except for the initial and final points. We show that corresponding to each path (\alpha: [0,1] \to \mathbb{R}^2) which connects (x) and (y) we have a zero in the path. Since there are infinitely many disjoint paths, we get infinitely many distinct zeros for (f).

Now, (\alpha: [0,1] \to \mathbb{R}^2) is a continuous function , ( \alpha(0)=x ), (\alpha(1)=y).

Consider the composition (g=f o \alpha : [0,1] \to \mathbb{R}). (g) is continuous. g(0)=f(x)>0 and g(1)=f(y)<0.

Therefore by the intermediate value theorem, (g(c)=0) for some (c\in [0,1]).

That means, (f(\alpha(c))=0). And using the discussion above we get a contradiction.

This proved the option A.


Chatuspathi

  • What is this topic: Real Analysis
  • What are some of the associated concept: Continuity,Intermediate Value Theorem
  • Book Suggestions: Introduction to Real Analysis by R.G. Bartle, D.R. Sherbert
Categories
College Mathematics

TIFR 2014 Problem 21 Solution – Determining function


TIFR 2014 Problem 21 Solution is a part of TIFR entrance preparation series. The Tata Institute of Fundamental Research is India’s premier institution for advanced research in Mathematics. The Institute runs a graduate programme leading to the award of Ph.D., Integrated M.Sc.-Ph.D. as well as M.Sc. degree in certain subjects.
The image is a front cover of a book named Introduction to Real Analysis by R.G. Bartle, D.R. Sherbert. This book is very useful for the preparation of TIFR Entrance.

Also Visit: College Mathematics Program of Cheenta


Problem:


Let (f:[0,1]\to [0,\infty ) ) be continuous. Suppose ( \int_{0}^{x} f(t)dt \ge f(x) ) for all (x\in [0,1]).

Then

A. no such function exists

B. there are infinitely many such functions

C. there is only one such function

D. there are exactly two such functions


Discussion:


Basically, the question is to find out how many such functions can exist.

Let (F(x)=\int_{0}^{x} f(t)dt ). That is, (F(x)) is the indefinite integral of (f).

We know from the fundamental theorem of calculus that:

(F'(x)=f(x)) .

So we have (F'(x) \ge F(x) ) for all (x\in [0,1] ).

We will manage this equation as we do in case of differential equations. Except that, we have an inequality here.

(F'(x)-F(x) \ge 0) for all (x\in [0,1] ).

Multiplying both sides by (e^{-x}) the inequality remain unchanged. This is because (e^{-x} >0 ).

(e^{-x}F'(x)-e^{-x}F(x) \ge 0) for all (x\in [0,1] ).

Now, ( (e^{-x}F(x))’ = e^{-x}F(x) -e^{-x}F'(x) ).

So we have (  (e^{-x}F(x))’ \le 0 ) for all (x\in [0,1] ).

Therefore, by taking integral from o to y and by using fundamental theorem of calculus:

(  e^{-y}F(y)-e^{0}F(0) \le 0 ) for all (y\in [0,1] ).

i.e, (  e^{-x}F(x) \le F(0) ) for all (x\in [0,1] ).

Also, note that by definition of (F), (F(0)=0). So we have

(  e^{-x}F(x) \le 0 ) for all (x\in [0,1] ).

But, since (e^{-x} >0 ) for all (x\in [0,1] ), we have (  F'(x) \le 0 )  for all (x\in [0,1] ).

So far, we have not used the fact that (f) is a non-negative function. Now we use it. Since (f(x) \ge 0) for all (x\in [0,1] ), therefore by monotonicity of the integral, (F) is an increasing function. This means (F'(x) \ge 0 ) for all (x\in [0,1] ).

By the two inequalities obtained above, we get (  F'(x) = 0 )  for all (x\in [0,1] ).

By the fundamental theorem (again!) we get (f(x)=F'(x)=0)  for all (x\in [0,1] ).

So there is only one such (f) namely the constant function (f=0).


Helpdesk

  • What is this topic: Real Analysis
  • What are some of the associated concept: Fundamental theorem of calculus, Increasing Function
  • Book Suggestions: Introduction to Real Analysis by R.G. Bartle, D.R. Sherbert
Categories
College Mathematics

TIFR 2014 Problem 20 Solution – Symmetries of Cube


TIFR 2014 Problem 20 Solution is a part of TIFR entrance preparation series. The Tata Institute of Fundamental Research is India’s premier institution for advanced research in Mathematics. The Institute runs a graduate programme leading to the award of Ph.D., Integrated M.Sc.-Ph.D. as well as M.Sc. degree in certain subjects.
The image is a front cover of a book named Abstract Algebra by David S. Dummit and Richard M. Foote. This book is very useful for the preparation of TIFR Entrance.

Also Visit: College Mathematics Program


Problem


Let (C) denote the cube ([-1,1]^3\subset \mathbb{R}^3 ). How many rotations are there in (\mathbb{R}^3) which take (C) to itself?


Discussion:


Let us label the six faces of the cube by (F_1,F_2,…,F_6).

Let (G) be the set consisting of all the rotations of (\mathbb{R}^3) which take (C) to itself. This set forms a group under the binary operation composition. In fact, (G) is a subgroup of the group of isometries of (\mathbb{R}^3). The identity is the identity transformation (rotation by 0 degree).

This group (G) acts on the set {(F_1,…F_6)}. The action is the most obvious one. A face (F_i) goes to face (F_j) under any rotation that preserves (C). That is, every rotation is a permutation of the faces (F_1,…,F_6). Said in other words: (\sigma . F_i = F_j) is the action where the face (F_i) goes to (F_j) under the rotation (\sigma).

We recall the Orbit-Stabilizer theorem:

( |G|=|orbit(x)||stab(x)| ). Where (G) acts on the set (X) and (x\in X).

Where (orbit(x)={ g.x | g\in G}) and (stab(x)={g\in G | g.x=x } ) is the stabilizer of (x). Here “.” denotes the action.

 

In this context, we have our group (G) as described above. And (X={F_1,…,F_6}).

In order to apply the orbit-stabilizer theorem, we look at (orbit(F_1)) and (stab(F_1)).

We can always rotate the cube in order that the face (F_1) goes to the face (F_2,..,F_6) and also the zero-rotation takes (F_1) to (F_1). Therefore (orbit(F_1)={F_1,…,F_6}).

What is the stabilizer of (F_1)? Well, the only rotations which fix the face (F_1) are the (0,\pi/2,2\pi/2,3\pi/2) rotations in the plane of (F_1) (i.e, having axis perpendicular to (F_1). Hence (|stab(F_1)|=4).

By orbit-stabilizer theorem we now have (|G|=6\times 4=24 ).


Helpdesk

  • What is this topic:Abstract Algebra
  • What are some of the associated concept: Orbit-Stabilizer Theorem
  • Book Suggestions: Abstract Algebra by David S. Dummit and Richard M. Foote.
Categories
College Mathematics

Last digit of \(97^{2013}\) (TIFR 2014 problem 18)

Question:

What is the last digit of (97^{2013})?

Discussion:

(97 \equiv -3 (\mod 10 ) )

(97^2 \equiv (-3)^2 \equiv -1 (\mod 10 ) )

(97^3 \equiv (-1)\times (-3) \equiv 3 (\mod 10 ) )

(97^4 \equiv (3)\times (-3) \equiv 1 (\mod 10 ) ).

Now, (2013=4\times 503 +1).

(97^{4\times 503+1} \equiv (1^{503})\times (97) \equiv 7 (\mod 10 ) ).

So the last digit of (97^{2013}) is 7.

Categories
College Mathematics

Subgroups of \(\mathbb{Z\times Z}\) (TIFR 2014 problem 16)

Question:

How many subgroups does the group (\mathbb{Z\times Z}) have?

A. 1

B. 2

C. 3

D. infinitely many.

Discussion:

How many subgroups does (\mathbb{Z}) have? Well, for every (m\ge 0) we have a subgroup (m\mathbb{Z}).

And these are all distinct. So there are infinitely many subgroups of  (\mathbb{Z}).

So how many subgroups does (\mathbb{Z}\times {\bar{0}}) have? Again, for every (m\ge 0) we have a subgroup (m\mathbb{Z}\times {\bar{0}}). That is again infinitely many.

Since (\mathbb{Z}\times {\bar{0}}) is a subgroup of  (\mathbb{Z\times Z}) we have infinitely many subgroups of (\mathbb{Z\times Z}).

Categories
College Mathematics

TIFR 2014 Problem 15 Solution -Discrete space


TIFR 2014 Problem 15 Solution is a part of TIFR entrance preparation series. The Tata Institute of Fundamental Research is India’s premier institution for advanced research in Mathematics. The Institute runs a graduate programme leading to the award of Ph.D., Integrated M.Sc.-Ph.D. as well as M.Sc. degree in certain subjects.
The image is a front cover of a book named Introduction to Real Analysis by R.G. Bartle, D.R. Sherbert. This book is very useful for the preparation of TIFR Entrance.

Also Visit: College Mathematics Program of Cheenta


Problem:


(X) is a metric space. (Y) is a closed subset of (X) such that the distance between any two points in (Y) is at most 1. Then

A. (Y) is compact.

B. any continuous function from (Y\to \mathbb{R}) is bounded.

C. (Y) is not an open subset of (X)

D. none of the above.


Discussion:


Let (X=) an infinite set for example (=\mathbb{R}) with the metric as discrete metric.

That is (d(x,y)=1) if (x\neq y) and (d(x,y)=0) if (x=y).

Then every set in (X) is open and every set is closed.

Now take (Y=X). Then (Y) can be covered by singleton sets. (Y=\cup {{a}|a\in Y}). Now each of the singleton sets is open in discrete metric space. Therefore, this is an open cover for (Y). Since (Y) is infinite, this cover has no finite subcover. So (Y) is not compact.

Given any (f:Y\to \mathbb{R}), for open set (U\in \mathbb{R}), (f^{-1}(U)\subset Y). Since Y is discrete, (f^{-1}(U))is open in (Y). So every function (f:Y\to \mathbb{R}) is a continuous function. In particular if we define (f(x)=x) then (f) is a continuous function. And (f) is not bounded.

Also, every subset of (X) is open. So (Y) is open.

Therefore, we are left with none of the above.


Helpdesk

  • What is this topic: Real Analysis
  • What are some of the associated concept: Continuous function, Discrete Metric Space, Finite subcover
  • Book Suggestions: Introduction to Real Analysis by R.G. Bartle, D.R. Sherbert