0% Complete
0/59 Steps

Connected Components: TIFR GS 2017 Part A Problem 7

Understand the problem

True or False? Let \(H_1, H_2, H_3,H_4 \) be four hyperplanes in \( R^3 \). The maximum possible number of connected components of \( R^3 – H_1 \cup H_2 \cup H_3 \cap H_4 \) is 14.
Source of the problem
TIFR GS 2017 Entrance Examination Paper
Topic
General Topology
Difficulty Level
Easy
Suggested Book
Topology, Second Edition, English, Paperback, by James R. Munkres

Start with hints

Do you really need a hint? Try it first!

If \(m\) is the number of hyperplanes and \(n\) is the dimension of the space in which the hyperplanes are intersecting then can you derive a formula which will give you the maximum possible number of connected components?
When \(m=2\) and \(n=1\) (that is, two hyperplanes are passing through a line, say) then the number of maximum possible connected components is \(2\), that is \({m \choose n} \).
When \(m=3\) and \(n=2\) (that is, three hyperplanes are intersecting in \(\Bbb R^2\) space) then the number of maximum possible connected components is \(3\), that is \({ m \choose n} \).
Hence, the maximum possible number of connected components when \(m\) hyperplanes are intersecting in \( \Bbb R^n\) is \( {m \choose 0}+{m \choose 1}+{m \choose 2}+{m \choose 3}+ \ldots +{m \choose n} \) when \( m>n\) and is \( 2^m \) when \(m<n\) and \(2^m-1\) when \(m=n+1 \).
Now, what do you think could be the maximum possible number of connected components in the given question?
The maximum possible number of connected components is \(15\). But the statement is saying \(14\). Hence the statement is false..

Watch the video (Coming Soon)

Connected Program at Cheenta

College Mathematics Program

The higher mathematics program caters to advanced college and university students. It is useful for I.S.I. M.Math Entrance, GRE Math Subject Test, TIFR Ph.D. Entrance, I.I.T. JAM. The program is problem driven. We work with candidates who have a deep love for mathematics. This program is also useful for adults continuing who wish to rediscover the world of mathematics.

Similar Problems

Application of eigenvalue in degree 3 polynomial: ISI MMA 2018 Question 14

This is a cute and interesting problem based on application of eigen values in 3 degree polynomial .Here we are finding the determinant value .

Order of rings: TIFR GS 2018 Part B Problem 12

This problem is a cute and simple application on the ring theory in the abstract algebra section. It appeared in TIFR GS 2018.

Last three digit of the last year: TIFR GS 2018 Part B Problem 9

This problem is a cute and simple application on the number theory in classical algebra portion. It appeared in TIFR GS 2018.

Group in graphs or graphs in groups ;): TIFR GS 2018 Part A Problem 24

This problem is a cute and simple application on the graphs in groups in the abstract algebra section. It appeared in TIFR GS 2018.

Problems on quadratic roots: ISI MMA 2018 Question 9

This problem is a cute and simple application on the problems on quadratic roots in classical algebra,. It appeared in TIFR GS 2018.

Are juniors countable if seniors are?: TIFR GS 2018 Part A Problem 21

This problem is a cute and simple application on the order of a countable groups in the abstract algebra section. It appeared in TIFR GS 2018.

Coloring problems: ISI MMA 2018 Question 10

This problem is a cute and simple application of the rule of product or multiplication principle in combinatorics,. It appeared in TIFR GS 2018.

Diagonilazibility in triangular matrix: TIFR GS 2018 Part A Problem 20

This problem is a cute and simple application on the diagonilazibility in triangular matrix in the abstract algebra section. It appeared in TIFR GS 2018.

Multiplicative group from fields: TIFR GS 2018 Part A Problem 17

This problem is a cute and simple application on the Multiplicative group from fields in the abstract algebra section. It appeared in TIFR GS 2018.

Group with Quotient : TIFR GS 2018 Part A Problem 16

This problem is a cute and simple application on Group theory in the abstract algebra section. It appeared in TIFR GS 2018.