Monthly Archives: August 2019

Set Theory, Relations, Functions Preliminaries: I

In these days of conflict between ancient and modern studies there must surely be something to be said of a study which did not begin with Pythagoras and will not end with Einstein. — G H Hardy (On Set Theory)

In every day life, we generally talk about group or collection of objects. Surely, you must have used the words such as team, bouquet, bunch, flock, family for collection of different objects.

It is very important to determine whether a given object belongs to a given collection or not. Consider the following conditions:

i) Successful persons in your city.

ii) Happy people in your town.

iii) Clever students in your class.

iv) Days in a week.

v) First five natural numbers.

Perhaps, you have already studied in earlier grade(s) —- can you state which of the above mentioned collections are sets? Why? Check whether your answers are as follows:

First three collections are not examples of sets but last two collections represent sets. This is because in first three collections, we are not sure of the objects. The terms ‘successful persons’, ‘happy people’, ‘clever students’ are all relative terms. Here, the objects are not well-defined. In the last two collections, we can determine the objects clearly (meaning, uniquely, or without ambiguity). Thus, we can say that the objects are well-defined.

So what can be the definition of a set ? Here it goes:

A collection of well-defined objects is called a set. (If we continue to “think deep” about this definition, we are led to the famous paradox, which Bertrand Russell had discovered: Let C be a collection of all sets such which are not elements of themselves. If C is allowed to be a set, a contradiction arises when one inquires whether or not C is an element of itself. Now plainly, there is something suspicious about the idea of a set being an element of itself, and we shall take this as evidence that the qualification “well-defined” needs to be taken seriously. Bertrand Russell re-stated this famous paradox in a very interesting way: In the town of Seville lives a barber who shaves everyone who does not shave himself. Does the barber shave himself?…)

The objects in a set are called elements or members of that set.

We denote sets by capital letters : A, B, C etc. The elements of a set are represented by small letters : a, b, c, d, e, f ….etc. If x is an element of a set A, we write x \in A. And, we read it as “x belongs to A.” If x is not an element of a set A, we write x \not\in A, and read as ‘x does not belong to A.’e.g., 1 is a “whole” number but not a “natural” number.

Hence, 0 \in W, where W is the set of whole numbers and 0 \not\in N, where N is a set of natural numbers.

There are two methods of representing a set:

a) Roster or Tabular Method or List Method (b) Set-Builder or Ruler Method

a) Roster or Tabular or List Method:

Let A be the set of all prime numbers less than 20. Can you enumerate all the elements of the set A? Are they as follows?

A=\{ 2,3,5,7,11,15,17,19\}

Can you describe the roster method? We can describe it as follows:

In the Roster method, we list all the elements of the set within braces \{, \} and separate the elements by commas.

In the following examples, state the sets using Roster method:

i) B is the set of all days in a week

ii) C is the set of all consonants in English alphabets.

iii) D is the set of first ten natural numbers.

2) Set-Builder Method:

Let P be the set of first five multiples of 10. Using Roster Method, you must have written the set as follows:

P = \{ 10, 20, 30, 40, 50\}

Question: What is the common property possessed by all the elements of the set P?

Answer: All the elements are multiples of 10.

Question: How many such elements are in the set?

Answer: There are 5 elements in the set.

Thus, the set P can be described using this common property. In such a case, we say that set-builder method is used to describe the set. So, to summarize:

In the set-builder method, we describe the elements of the set by specifying the property which determines the elements of the set uniquely.

Thus, we can write : P = \{ x: x =10n, n \in N, n \leq 5\}

In the following examples, state the sets using set-builder method:

i) Y is the set of all months of a year

ii) M is the set of all natural numbers

iii) B is the set of perfect squares of natural numbers.

Also, if elements of a set are repeated, they are written once only; while listing the elements of a set, the order in which the elements are listed is immaterial. (but this situation changes when we consider sets from the view-point of permutations and combinations. Just be alert in set-theoretic questions.)

Subset: A set A is said to be a subset of a set B if each element of set A is an element of set B. Symbolically, A \subseteq B.

Superset: If A \subset B, then B is called the superset of set A. Symbolically: B \supset A

Proper Subset: A non empty set A is said to be a proper subset of the set B, if and only if all elements of set A are in set B, and at least one element of B is not in A. That is, if A \subseteq B, but A \neq B then A is called a proper subset of B and we write A \subset B.

Note: the notations of subset and proper subset differ from author to author, text to text or mathematician to mathematician. These notations are not universal conventions in math.

Intervals: 

  1. Open Interval : given a < b, a, b \in R, we say a<x<b is an open interval in \Re^{1}.
  2. Closed Interval : given a \leq x \leq b = [a,b]
  3. Half-open, half-closed: a <x \leq b = (a,b], or a \leq x <b=[a,b)
  4. The set of all real numbers greater than or equal to a : x \geq a =[a, \infty)
  5. The set of all real numbers less than or equal to a is (-\infty, a] = x \leq a

Types of Sets:

  1. Empty Set: A set containing no element is called the empty set or the null set and is denoted by the symbol \phi or \{ \} or void set. e.g., A= \{ x: x \in N, 1<x<2\}
  2. Singleton Set: A set containing only one element is called a singleton set. Example : (i) Let A be a set of all integers which are neither positive nor negative. Then, A = \{ 0\} and example (ii) Let B be a set of capital of India. Then B= \{ Delhi\}

We will define the following sets later (after we giving a working definition of a function): finite set, countable set, infinite set, uncountable set.

3. Equal sets: Two sets are said to be equal if they contain the same elements, that is, if A \subseteq B and B \subseteq A. For example: Let X be the set of letters in the word ‘ABBA’ and Y be the set of letters in the word ‘BABA’. Then, X= \{ A,B\} and Y= \{ B,A\}. Thus, the sets X=Y are equal sets and we denote it by X=Y.

How to prove that two sets are equal?

Let us say we are given the task to prove that A=B, where A and B are non-empty sets. The following are the steps of the proof : (i) TPT: A \subset B, that is, choose any arbitrary element x \in A and show that also x \in B holds true. (ii) TPT: B \subset A, that is, choose any arbitrary element y \in B, and show that also y \in A. (Note: after we learn types of functions, we will see that a fundamental way to prove two sets (finite) are equal is to show/find a bijection between the two sets).

PS: Note that two sets are equal if and only if they contain the same number of elements, and the same elements. (irrespective of order of elements; once again, the order condition is changed for permutation sets; just be alert what type of set theoretic question you are dealing with and if order is important in that set. At least, for our introduction here, order of elements of a set is not important).

PS: Digress: How to prove that in general, x=y? The standard way is similar to above approach: (i) TPT: x < y (ii) TPT: y < x. Both (i) and (ii) together imply that x=y.

4. Equivalent sets: Two finite sets A and B are said to be equivalent if n(A)=n(B). Equal sets are always equivalent but equivalent sets need not be equal. For example, let A= \{ 1,2,3 \} and B = \{ 4,5,6\}. Then, n(A) = n(B), so A and B are equivalent. Clearly, A \neq B. Thus, A and B are equivalent but not equal.

5. Universal Set: If in a particular discussion all sets under consideration are subsets of a set, say U, then U is called the universal set for that discussion. You know that the set of natural numbers the set of integers are subsets of set of real numbers R. Thus, for this discussion is a universal set. In general, universal set is denoted by or X.

6. Venn Diagram: The pictorial representation of a set is called Venn diagram. Generally, a closed geometrical figures are used to represent the set, like a circle, triangle or a rectangle which are known as Venn diagrams and are named after the English logician John Venn.

In Venn diagram the elements of the sets are shown in their respective figures.

Now, we have these “abstract toys or abstract building-blocks”, how can we get new such “abstract buildings” using these “abstract building blocks”. What I mean is that we know that if we are a set of numbers like 1,2,3, …, we know how to get “new numbers” out of these by “adding”, subtracting”, “multiplying” or “dividing” the given “building blocks like 1, 2…”. So, also what we want to do now is “operations on sets” so that we create new, more interesting or perhaps, more “useful” sets out of given sets. We define the following operations on sets:

  1. Complement of a set: If A is a subset of the universal set U then the set of all elements in U which are not in A is called the complement of the set A and is denoted by A^{'} or A^{c} or \overline{A} Some properties of complements: (i) {A^{'}}^{'}=A (ii) \phi^{'}=U, where U is universal set (iii) U^{'}= \phi
  2. Union of Sets: If A and B are two sets then union of set A and set B is the set of all elements which are in set A or set B or both set A and set B. (this is the INCLUSIVE OR in digital logic) and the symbol is : $latex A \bigcup B
  3. Intersection of sets: If A and B are two sets, then the intersection of set A and set B is the set of all elements which are both in A and B. The symbol is A \bigcap B.
  4. Disjoint Sets: Let there be two sets A and B such that A \bigcap B=\phi. We say that the sets A and B are disjoint, meaning that they do not have any elements in common. It is possible that there are more than two sets A_{1}, A_{2}, \ldots A_{n} such that when we take any two distinct sets A_{i} and A_{j} (so that i \neq j, then A_{i}\bigcap A_{j}= \phi. We call such sets pairwise mutually disjoint. Also, in case if such a collection of sets also has the property that \bigcup_{i=1}^{i=n}A_{i}=U, where U is the Universal Set in the given context, We then say that this collection of sets forms a partition of the Universal Set.
  5. Difference of Sets: Let us say that given a universal set U and two other sets A and B, B-A denotes the set of elements in B which are not in A; if you notice, this is almost same as A^{'}=U-A.
  6. Symmetric Difference of Sets: Suppose again that we are two given sets A and B, and a Universal Set U, by symmetric difference of A and B, we mean (A-B)\bigcup (B-A). The symbol is A \triangle B. Try to visualize this (and describe it) using a Venn Diagram. You will like it very much. Remark : The designation “symmetric difference” for the set A \triangle B is not too apt, since A \triangle B has much in common with the sum A \bigcup B. In fact, in A \bigcup B the statements “x belongs to A” and “x belongs to B” are joined by the conjunction “or” used in the “either …or …or both…” sense, while in A \triangle B the same two statements are joined by “or” used in the ordinary “either…or….” sense (as in “to be or not to be”). In other words, x belongs to A \bigcup B if and only if x belongs to either A or B or both, while x belongs to A \triangle B if and only if x belongs to either A or B but not both. The set A \triangle B can be regarded as a kind of a “modulo-two-sum” of the sets A and B, that is, a sum of the sets A and B in which elements are dropped if they are counted twice (once in A and once in B).

Let us now present some (easily provable/verifiable) properties of sets:

  1. A \bigcup B = B \bigcup A (union of sets is commutative)
  2. (A \bigcup B) \bigcup C = A \bigcup (B \bigcup C) (union of sets is associative)
  3. A \bigcup \phi=A
  4. A \bigcup A = A
  5. A \bigcup A^{'}=U where U is universal set
  6. If A \subseteq B, then A \bigcup B=B
  7. U \bigcup A=U
  8. A \subseteq (A \bigcup B) and also B \subseteq (A \bigcup B)

Similarly, some easily verifiable properties of set intersection are:

  1. A \bigcap B = B \bigcap A (set intersection is commutative)
  2. (A \bigcap B) \bigcap C = A \bigcap (B \bigcap C) (set intersection is associative)
  3. A \bigcap \phi = \phi \bigcap A= \phi (this matches intuition: there is nothing common in between a non empty set and an empty set :-))
  4. A \bigcap A =A (Idempotent law): this definition carries over to square matrices: if a square matrix is such that A^{2}=A, then A is called an Idempotent matrix.
  5. A \bigcap A^{'}=\phi (this matches intuition: there is nothing in common between a set and another set which does not contain any element of it (the former set))
  6. If A \subseteq B, then A \bigcap B =A
  7. U \bigcap A=A, where U is universal set
  8. (A \bigcap B) \subseteq A and (A \bigcap B) \subseteq B
  9. i: A \bigcap (B \bigcap )C = (A \bigcap B)\bigcup (A \bigcap C) (intersection distributes over union) ; (9ii) A \bigcup (B \bigcap C)=(A \bigcup B) \bigcap (A \bigcup C) (union distributes over intersection). These are the two famous distributive laws.

The famous De Morgan’s Laws for two sets are as follows: (it can be easily verified by Venn Diagram):

For any two sets A and B, the following holds:

i) (A \bigcup B)^{'}=A^{'}\bigcap B^{'}. In words, it can be captured beautifully: the complement of union is intersection of complements.

ii) (A \bigcap B)^{'}=A^{'} \bigcup B^{'}. In words, it can be captured beautifully: the complement of intersection is union of complements.

Cardinality of a set: (Finite Set) : (Again, we will define the term ‘finite set’ rigorously later) The cardinality of a set is the number of distinct elements contained in a finite set A and we will denote it as n(A).

Inclusion Exclusion Principle:

For two sets A and B, given a universal set U: n(A \bigcup B) = n(A) + n(B) - n(A \bigcap B).

For three sets A, B and C, given a universal set U: n(A \bigcup B \bigcup C)=n(A) + n(B) + n(C) -n(A \bigcap B) -n(B \bigcap C) -n(C \bigcup A) + n(A \bigcap B \bigcap C).

Homework Quiz: Verify the above using Venn Diagrams. 

Power Set of a Set:

Let us consider a set A (given a Universal Set U). Then, the power set of A is the set consisting of all possible subsets of set A. (Note that an empty is also a subset of A and that set A is a subset of A itself). It can be easily seen (using basic definition of combinations) that if n(A)=p, then n(power set A) = 2^{p}. Symbol: P(A).

Homework Tutorial I:

  1. Describe the following sets in Roster form: (i) \{ x: x \hspace{0.1in} is \hspace{0.1in} a \hspace{0.1in} letter \hspace{0.1in} of \hspace{0.1in} the \hspace{0.1in} word \hspace{0.1in}  PULCHRITUDE\} (II) \{ x: x \hspace{0.1in } is \hspace{0.1in} an \hspace{0.1in} integer \hspace{0.1in} with \hspace{0.1in} \frac{-1}{2} < x < \frac{1}{2} \} (iii) \{x: x=2n, n \in N\}
  2. Describe the following sets in Set Builder form: (i) \{ 0\} (ii) \{ 0, \pm 1, \pm 2, \pm 3\} (iii) \{ \}
  3. If A= \{ x: 6x^{2}+x-15=0\} and B= \{ x: 2x^{2}-5x-3=0\}, and x: 2x^{2}-x-3=0, then find (i) A \bigcup B \bigcup C (ii) A \bigcap B \bigcap C
  4. If A, B, C are the sets of the letters in the words, ‘college’, ‘marriage’, and ‘luggage’ respectively, then verify that \{ A-(B \bigcup C)\}= \{ (A-B) \bigcap (A-C)\}
  5. If A= \{ 1,2,3,4\}, B= \{ 3,4,5, 6\}, C= \{ 4,5,6,7,8\} and universal set X= \{ 1,2,3,4,5,6,7,8,9,10\}, then verify the following:

5i) A\bigcup (B \bigcap C) = (A\bigcup B) \bigcap (A \bigcup C)

5ii) A \bigcap (B \bigcup C)= (A \bigcap B) \bigcup (A \bigcap C)

5iii) A= (A \bigcap B)\bigcup (A \bigcap B^{'})

5iv) B=(A \bigcap B)\bigcup (A^{'} \bigcap B)

5v) n(A \bigcup B)= n(A)+n(B)-n(A \bigcap B)

6. If A and B are subsets of the universal set is X, n(X)=50, n(A)=35, n(B)=20, n(A^{'} \bigcap B^{'})=5, find (i) n(A \bigcup B) (ii) n(A \bigcap B) (iii) n(A^{'} \bigcap B) (iv) n(A \bigcap B^{'})

7. In a class of 200 students who appeared certain examinations, 35 students failed in MHTCET, 40 in AIEEE, and 40 in IITJEE entrance, 20 failed in MHTCET and AIEEE, 17 in AIEEE and IITJEE entrance, 15 in MHTCET and IITJEE entrance exam and 5 failed in all three examinations. Find how many students (a) did not flunk in any examination (b) failed in AIEEE or IITJEE entrance.

8. From amongst 2000 literate and illiterate individuals of a town, 70 percent read Marathi newspaper, 50 percent read English newspapers, and 32.5 percent read both Marathi and English newspapers. Find the number of individuals who read

8i) at least one of the newspapers

8ii) neither Marathi and English newspaper

8iii) only one of the newspapers

9) In a hostel, 25 students take tea, 20 students take coffee, 15 students take milk, 10 students take both tea and coffee, 8 students take both milk and coffee. None of them take the tea and milk both and everyone takes at least one beverage, find the number of students in the hostel.

10) There are 260 persons with a skin disorder. If 150 had been exposed to chemical A, 74 to chemical B, and 36 to both chemicals A and B, find the number of persons exposed to  (a) Chemical A but not Chemical B (b) Chemical B but not Chemical A (c) Chemical A or Chemical B.

11) If A = \{ 1,2,3\} write down the power set of A.

12) Write the following intervals in Set Builder Form: (a) (-3,0) (b) [6,12] (c) (6,12] (d) [-23,5)

13) Using Venn Diagrams, represent (a) (A \bigcup B)^{'} (b) A^{'} \bigcup B^{'} (c) A^{'} \bigcap B (d) A \bigcap B^{'}

Regards,

Nalin Pithwa.

IITJEE Foundation Maths: Variation

DEFINITION:

One quantity A is said to vary directly as another B, when the two quantities depend upon each other in such a manner that if B is changed, A is changed in the same ratio.

NOTE: The word directly is often omitted, and A is said to vary as B.

For instance: if a train a moving at a uniform rate travels 40 miles in 60 minutes, it will travel 20 miles in 30 minutes, 80 miles in 120 minutes, and so on; the distance in each case being increased or diminished in the same ratio as the time. This is expressed by saying that when the velocity is uniform the distance is proportional to the time, or the distance varies as the time.

NOTATION: The symbol \alpha is used to denote variation; so that, A \alpha B is read as “A varies as B.”

Theorem I: If A varies as B, then A is equal to B multiplied by some constant quantity.

Note: If any pair of corresponding values of A and B are known, the constant m can be determined. For instance, if A=B, when B=12, we have 3=m \times 12; and m=\frac{1}{4}, and A=\frac{1}{4}B

DEFINITION: One quantity A is said to vary inversely as another B, when A varies directly as the reciprocal of B.

The following is an illustration of inverse variation: If 6 men do a certain work in 8 hours, 12 men would do the same work in 4 hours; 2 men in 24 hours; and so on. Thus, it appears that when the number of men is increased, the same is proportionately decreased; and vice-versa.

Example 1: The cube root of x varies inversely as the square of y; if x=8, when y=3; find x when y=\frac{3}{2}.

Solution 1: By supposition, \sqrt[3]{x}=\frac{m}{y^{2}}, where m is constant. Putting x=8, y=3, we have 2=\frac{m}{9}, so m=18, and \sqrt[3]{x}=\frac{18}{y^{2}}; hence, by putting y=\frac{3}{2}, we obtain x=512.

Example 2: The square of the time of a planet’s revolution varies as the cube of its distance from the Sun; find the time of Venus’s revolution, assuming the distances of the Earth and Venus from the Sun to be 91\frac{1}{4} and 66 millions of miles respectively.

Let P be the periodic time measured in days, D the distance in millions of miles; we have

P^{2} \alpha D^{3}, or P^{2}=k \times D^{3}, where k is some constant.

For the Earth, 365 \times 365 = k \times 91 \frac{1}{4} \times 91 \frac{1}{4} \times 91 \times \frac{1}{4}

hence, k=\frac{4 \times 4 \times 4}{3.5}

so that P^{2}=\frac{4 \times 4 \times 4}{365}D^{3}

For Venus, P^{2}=\frac{4 \times 4 \times 4}{365} \times 66 \times 66 \times 66

hence, P= A \times 66 \times \sqrt{\frac{264}{365}} = 264 \times \sqrt{0.7233}, approximately.

P^{2}=264 \times 0.85=224.4.

Hence, the time of revolution is nearly 224\frac{1}{2}.

DEFINITION: One quantity is said to vary jointly as a number of others, when it varies directly as their product.

Thus, A varies jointly as B and C, when A=m \times BC. For instance, the interest on a sum of money varies jointly as the principal, the time, and the rate per cent.

DEFINITION: 

A is said to vary directly as B and inversely as C, when A varies as \frac{B}{C}.

Theorem:

If A varies as B, when C is constant, and A varies as C when B is constant, then A will vary as BC when both B and C vary.

Proof:

Homework

The following are some illustrations of the theorems stated above:

The amount of work done by a given number of men varies directly as the number of days they work, and the amount of work done in a given time varies directly as the number of men; therefore, when the number of days and the number of men are both variable, the amount of work will vary as the product of the number of men and the number of days.

Again, in plane geometry, the area of a triangle varies directly as its base when the height is constant, and directly as the height when the base is constant; and when both the height and base are variable, the area varies as the product of the numbers representing the height and the base.

Example:

The volume of a right circular cone varies as the square of the radius of the base when the height is constant, and as the height when the base is constant. If the radius of base is 7cm, and the height is 15 cm, the volume is 770 cc, find the height of a cone whose volume is 132 cubic cm, and which stands on a base whose radius is 3cm.

Solution:9 

Let h and r denote respectively the height and radius of the base measured in cm.; also let V be the volume in cubic cm.

Then. V=m \times r^{2} \times h, where m is constant.

By assumption, 770=m \times 7^{2} \times 15

hence, m = \frac{22}{21} so that V=\frac{22}{21}r^{2}h.

By substituting V=132, r=3, we get the following:

132=\frac{22}{21} \times 9 \times h

so that h=14; and, therefore the height is 14 cm.

Note:

A quantity A can vary jointly as (a product of) more than two variables also as is most often the case in real engineering. Further, the variations may be either direct or inverse. The principle is interesting because of its frequent occurence in physical sciences or engineering. For example, Boyle’s law in chemistry: It is found by experiment that the pressure (P) of a gas varies as the “absolute temperature” (T) (in Kelvin) when its volume (V) is constant and that the pressure varies inversely as the volume when the temperature is constant; that is,

P \alpha T, when V is constant and P \alpha \frac{1}{V} when T is constant.

From these results we should expect that, when both t and v are variable, we should have the formula:

P \alpha \frac{T}{V}, or PV=kT, where k is a constant (based on laws of chemistry). And, by actual experiment this is found to be true.

Example:

The duration of a railway journey varies ditectly as the distance and inversely as the velocity; the velocity varies directly as the square root of the quantity of coal used per kilometer (don’t worry the days of steam engine/coal engine and resulting environmental degradation are over; but this is only a simple engineering application), and inversely as the number of carriages in the train. In a journey of 50 kilometers, in half an hour with 18 carriages 100 kg of coal is required; how much coal will be consumed in a journey of 42 kilometers, in 28 minutes with 16 carriages?

Solution:

Let t be the time expressed in hours; let d be the distance in kilometers; let v be the velocity in kmph; let q be the mass of coal (in kg) used per kilometers; and let c be the number of carriages.

We have t \alpha \frac{d}{v} and v \alpha \frac{\sqrt{q}}{c}, and hence, t \alpha \frac{cd}{\sqrt{q}}, or t=\frac{kcd}{\sqrt{q}}, where k is a constant.

Substituting the values given, we have (since q=2),

\frac{1}{2} = \frac{k \times 18 \times 50}{\sqrt{2}}

that is, k=\frac{1}{\sqrt{2} \times 18 \times 50}.

Hence, t=\frac{cd}{\sqrt{2} \times 18 \times 50\sqrt{q}}

Substituting now the values of t, c, d given in the second part of the question, we have

\frac{28}{60}=\frac{16 \times 42}{\sqrt{2} \times 18 \times 50 \times \sqrt{q} }

that is, \sqrt{q} = 4 \sqrt{2}, hence q=32.

Hence, the quantity of coal is 42 \times 32 = 1344 kg.

Tutorial problems on Variation:

  1. If x varies as y, and x=8, when y=15, find x when y=10
  2. If P varies as Q, and P=7 when Q=3, find P when Q=2\frac{1}{3}.
  3.  If the square of x varies as the cube of y, and x=3, when y=4, find the value of y when x=\frac{1}{\sqrt{3}}.
  4. A varies as B and C jointly; if A=2 when B=\frac{3}{5} and C=\frac{10}{27}, find C when A=54 and B=3.
  5. If A varies as C, and B varies as C, then A \pm B and \sqrt{AB} will each vary as C.
  6. If A varies as BC, then B varies inversely as \frac{C}{A}.
  7. P varies directly as Q and inversely as R; also P=\frac{2}{3} when Q=\frac{3}{7} and R=\frac{9}{14}; find Q when P=\sqrt{48} and R=\sqrt{75}.
  8. If x varies as y, prove that x^{2}+y^{2} varies as x^{2}-y^{2}.
  9. If y varies as the sum of two quantities, of which one varies directly as x and the other inversely as x; and if y=6 then x=4, and y=3\frac{1}{3} when x=3, find the equation between x and y.
  10. If y is equal to the sum of two quantities one of which varies as x directly, and the other as x^{2} inversely; and, if y=19 when x=2, or 3; find y in terms of x.
  11. If A varies directly as the square root of B and inversely as the cube of C, and if A=3, when B=256 and C=2, find B when A=24 and C = \frac{1}{2}
  12. Given that x+y varies as x+\frac{1}{x}, and that x-y varies as z- \frac{1}{z}, find the relation between x and z, provided that z=2 when x=3 and y=1.
  13. If A varies as B and C jointly, write B varies as D^{2}, and C varies inversely as A, show that A varies as D.
  14. If y varies as the sum of three quantities of which the first is a constant, the second varies as x, and the third as x^{2}; and, if y=0 when x=1, y=1, when x=2, and y=4 when x=3; find y when x=7.
  15. When a body falls down from rest the distance from the starting point varies as the square of the time it has been falling; if a body falls through 122.6 meters in 5 seconds, how far does it fall in 10 seconds? Also, how far does it fall in the tenth second?
  16. Given that the volume of a sphere varies as the cube of its radius, and that when the radius is 3.5 cm, the volume is 176.7 cubic cm; find the volume when the radius is 1.75 cm.
  17. The weight of a circular disc varies as the square of the radius when the thickness remains the same; it also varies as the thickness when the radius remains the same. Two discs have their thicknesses in the ratio of 9:8; find the ratio of their radii if the weight of the first is twice that of the second.
  18. At a certain regatta, the numbers of races on each day varied jointly as the number of days from the beginning and end of the regatta up to and including the day in question. On three successive days there were respectively 6, 5 and 3 races. Which days were these, and how long did the regatta last?
  19. The price of a diamond varies as the square of its weight (mass). Three rings of equal weight, each composed of a diamond set in gold, have values INR a, INR b, INR c, the diamonds in them weighing 3, 4, 5 carats respectively. Show that the value of a diamond of one carat is INR (\frac{a+c}{2}-b), the cost of workmanship being the same for each ring.
  20. Two persons are awarded pensions in proportion to the square root of the number of root of the number of years they have served. One has served 9 years longer than the other and receives a pension greater by INR 500. If the length of service of the first had exceeded that of the second by 4\frac{1}{4} years their pensions would have been in the proportion of 9:8. How long had they served and what were their respective pensions?
  21. The attraction of a planet on the satellites varies directly as the mass (M) of the planet, and inversely as the square of the distance (D); also the square of a satellite’s time of revolution varies directly as the distance and inversely as the force of attraction. If m_{1}, d_{1}, t_{1} and m_{2}, d_{2}, t_{2} are simultaneous values of M, D, T respectively, prove that \frac{m_{1}t_{1}^{2}}{m_{2}t_{2}^{2}} = \frac{d_{1}^{3}}{d_{2}^{3}}. Hence, find the time of revolution of that moon of Jupiter whose distance is to the distance of our Moon as 35:31, having given that the mass of Jupiter is 343 times that of the Earth, and that the Moon’s period is 27.32 days.
  22. The consumption of coal by a locomotive varies as the square of the velocity; when the speed is 32 kmph the consumption of coal per hour is 2 tonnes: if the price of coal is INR 10 per tonne, and the other expenses of the engine be INR 11.25 an hour, find the least cost of a journey of 100 km.

Cheers,

Nalin Pithwa

 

 

 

 

 

 

 

 

 

 

Two powerful wise quotes

  1. I am a pessimist by logic, but optimist by will-power. — Anon.
  2. The only thing greater than the power of the human mind is the courage of the human heart. — John Forbes Nash, Jr., Nobel Laureate mathematician (Economics Prize), Abel Laureate, victim of paranoid schizophrenia for 30 years.

 

Some fun – Math Late Show with David Letterman and Daniel Tammet

Stretching is a good exercise but…

Stretching is a good exercise, but stretching the mind through math is even better !!

Ha…ha…ha…LOL 🙂

Nalin Pithwa

PS: An object does not change topologically if it is “stretched”…Ha, ha, ha…LOL 🙂

Games, social behavior, chess, economics and maths

The following is some trivia but in fact, not so trivia, in this age of data science, data analytics, social media platforms, on-line gaming etc…If you decide to ponder over deep…you will become a giant mathematician or applied mathematician or of course, a computer science wunderkind..

The following is “picked out as it is” from a famous biography, (which regular readers of my blog will now know, perhaps, is a favorite mathematical biography for me)…A Beautiful Mind by Sylvia Nasar, biography of mathematical genius, John Forbes Nash, Jr, Nobel Laureate (Economics) and Abel Laureate:

“It was the great Hungarian-born polymath John von Neumann who first recognized that social behaviour could be analyzed as games. Von Neumann’s 1928 article on parlor games was the first successful attempt to derive logical and mathematical rules about rivalries. Just as William Blake saw the universe in a grain of sand, great scientists have often looked for clues in vast and complex problems in the small, familiar phenomena of daily life. Isaac Newton reached insights about the heavens by juggling wooden balls. Albert Einstein contemplated a boat paddling upriver. Von Neumann pondered the game of poker.

A seemingly trivial and playful pursuit like poker, von Neumann argued, might hold the key to more serious human affairs for two reasons. Both poker and economic competition require a certain type of reasoning, namely the rational calculation of advantage and disadvantage based on some internally consistent system of values (“more is better than less’). And, in both, the outcome for any individual actor depends not only on his own actions, but on the independent actions of others.

More than a century earlier, the French economist Antoine-Augustin Cournot had pointed out that problems of economic cnoice were greatly simplified when either none or a large number of other agents were present. Alone on his island, Robinson Crusoe does not have to worry about whose actions might affect him. Neither do Adam Smith’s butchers and bakers. They live in a world with so many others that their actions, in effect, cancel each other out. But when there is more than one agent but not so many that their influence may be safely ignored, strategic behavior raises a seemingly insoluble problem:”I think that he thinks that I think that he thingks,” and so forth…

So play games but think math ! 🙂

Nalin Pithwa

 

 

You and your research ( You and your studies) : By Richard Hamming, AT and T, Bell Labs mathematician;

Although the title is grand (and quite aptly so)…the reality is that it can be applied to serious studies for IITJEE entrance, CMI entrance, highly competitive math olympiads, and also competitive coding contests…in fact, to various aspects of student life and various professional lifes…

Please read the whole article…apply it wholly or partially…modified or unmodified to your studies/research/profession…these are broad principles of success…

https://www.cs.virginia.edu/~robins/YouAndYourResearch.html

 

Tricky Trigonometry questions for IITJEE mains maths practice

Prove the following:

  1. \frac{\sin{A}}{1-\cos{A}} = \frac{1+\cos{A}-\sin{A}}{\sin{A}-1+\cos{A}}
  2. \frac{1+\sin{A}}{\cos{A}}=\frac{1+\sin{A}+\cos{A}}{\cos{A}+1-\sin{A}}
  3. \frac{\tan{A}}{\sec{A}-1}=\frac{\tan{A}+\sec{A}+1}{\sec{A}-1+\tan{A}}
  4. \frac{1+\csc{A}+ \cot{A}}{1+\csc{A}-\cot{A}} = \frac{\csc{A}+\cot{A}-1}{\cot{A}-\csc{A}+1}

Hint: Directly trying to prove LHS is RHS is difficult in all the above; or even trying to transform RHS to LHS is equally difficult; it is quite easier to prove the equivalent statement by taking cross-multiplication of the appropriate expressions. 🙂

Regards,

Nalin Pithwa.

What motivated Einstein?

The most beautiful thing that we can experience is the mysterious. It is the source of all true art and sciences.

— Albert Einstein, in What I believe, 1930.

E. T. Bell’s Men of Mathematics, John Nash, Jr., genius mathematician, Nobel Laureate and Abel Laureate; and Albert Einstein

(From A Beautiful Mind by Sylvia Nasar)

The first bite of mathematical apple probably occurred when Nash at around age thirteen or fourteen read E. T. Bell’s extra ordinary book Men of Mathematics — an experience he alludes to in his autobiographical essay (of Nobel Prize, Economics) Bell’s book, which was published in 1937, would have given Nash the first glimpse of real mathematics, a heady realm of symbols and mysteries entirely unconnected to the seemingly arbitrary and dull rules of arithmetic and geometry taught in school or even in the entertaining but ultimately trivial calculations that Nash carried out in the course of chemistry and electrical experiments.

Men of Mathematics consists of lively — and, as it turns out, not entirely accurate — biographical sketches. Its flamboyant author, a professor of mathematics at California Institute of Technology, declared himself disgusted with “the ludicrous untruth of the traditional portrait of the mathematician” as a “slovenly dreamer totally devoid of common sense.” He assured his readers that the great mathematicians of history were an exceptionally virile and even adventuresome breed. He sought to prove his point with vivid accounts of infant precocity, monstrously insensitive educational authorities, crushing poverty, jealous rivals, love affairs, royal patronage, and many varieties of early death, including some resulting from duels. He even went so far in defending mathematicians as to answer the question : “How many of the great mathematicians have been perverts?” None, was his answer. ‘Some lived celibate lives, usually on account of economic disabilities, but the majority were happily married…The only mathematician discussed here whose life might offer something of interest to a Freudian is Pascal.’ The book became a bestseller as soon as it appeared.

What makes Bell’s account not merely charming, but intellectually seductive, are his lively descriptions of mathematical problems that inspired his subjects when they were young, and his breezy assurance that there were still deep and beautiful problems that could be solved by amateurs, boys of fourteen, to be specific. It was Bell’s essay on Fermat, one of the greatest mathematicians of all time, but a perfectly conventional seventeenth century French magistrate, whose life was “quiet, laborious and uneventful,” that caught Nash’s eye. The main interest of Fermat, who shares the credit for inventing calculus with Newton and analytic geometry with Descartes, was number theory — “the higher arithmetic.” Number theory, investigates the natural relationships of those common whole numbers 1, 2, 3, 4, 5…which we utter almost as soon as we learn to talk.

For Nash, proving a theorem known as Fermat’s (Little) Theorem about prime numbers, those mysterious integers that have no divisors besides themselves and one produced an epiphany of sorts. Often mathematical geniuses, Albert Einstein and Bertrand Russell among them recount similar revelatory experiences in early adolescence. Einstein recalled the “wonder” of his first encounter with Euclid at age twelve:

“Here were assertions, as for example the intersection of three altitudes of a triangle at one point which — though by no means evident — could nevertheless be proved with such certainty that any doubt appeared to be out of the question. This lucidity and certainty made an indescribable impression on me.”

Nash does not describe his feelings when he succeeded in devising a proof for Fermat’s assertion that if n is any whole number and p any prime number, then n multiplied by itself p times minus p is divisible by p. But, he notes the fact in his autobiographical essay, and his emphasis on this concrete result of his initial encounter with Fermat suggests that the thrill of discovering and exercising his own intellectual powers — as much as any sense of wonder inspired by hitherto unsuspected patterns and meanings — was what made this moment such a memorable one. That thrill has been decisive for many a future mathematician. Bell describes how success in solving a problem posed by Fermat led Carl Friedrich Gauss, the renowned German mathematician, to choose between two careers for which he was similarly talented. ‘It was this discovery …which induced the young man to choose mathematics instead of philology as his life work.”…

For those readers who are interested:

  1. Who wants to be a mathematician:

http://www.ams.org/publicoutreach/students/wwtbam/wwtbam

2. Resonance Journal (India):

https://www.ias.ac.in/Journals/Resonance_%E2%80%93_Journal_of_Science_Education/

3. Ramanujan School of Mathematics; Super30 of Prof Anand Kumar:

http://www.super30.org/rsm.html

Cheers,

Nalin Pithwa