why is an empty set a subset of itself

Note first of all the definition of a subset: We say A \subset B only when: if x \in A then x \in B.

The empty set is the set with no elements. It is denoted by the symbol \phi. A source of confusion to beginners is that although the empty set consists of nothing, it itself is something (namely, some particular set, the one characterized by the fact that nothing is in it). The set \{\phi \} is a set containing exactly one element, namely the empty set. (In a similar way, when dealing with numbers, say with ordinary integers, we must be careful not to regard the number zero as nothing; zero is something, a particular number, which represents the numbers in “nothing”. Thus, zero and \phi are quite different, but there is a connection between them in that the set has zero elements.) Note that for any set X, we have

\phi \subset X and X \subset X.

A special case of both of these statements is the statement  which occasions difficulty if, as is often improperly done, one reads “is contained in” for both of the symbols \phi and \in The statement

\phi \subset \phi

is true because the statement “for each x \in \phi, we have x \in \phi” is obviously true, and also because it is “vacuously true”, that is there is no for which the statement must be verified, just as the statement “all pigs with wings speak Chinese” is vacuously true.

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Do send your questions, suggestions, comments, more such seemingly paradoxical statements to me.

More later, happy new year 🙂

Nalin Pithwa

What is Madhava Mathematics Competition

This blog is especially for the Madhava Mathematics Competition and IITJEE Main and Advanced mathematics and also BSc Pure Mathematics of India.

Official information about the Madhava Mathematics Competition can be found at its official website:

http://www.madhavacompetition.com/

It is conducted by Department of Mathematics, S.P. College, Pune and TIFR (Tata Institute of  Fundamental Research)/Homibhabha Centre for Science Education. IG is like “baby Putnam” , to use the analogy of William Lowell Putnam competition of the USA.

(A Mathematics Competition for Undergraduate Students)

The competition is named after Madhava, who introduced in the fourteenth centurry, profound mathematical ideas that are now part of Calculus. His most famous achivements include the Madhava – Leibnitz series for π, the Madhava – Newton power series for sine and Cosine functions and a numerical value of π that is accurate to eleven decimal places. The “Madhava School”, consisting of a long chain of teachers-students continuity, flourished for well over two centuries, making significant contributions to mathematics and astronomy.

MADHAVA MATHEMATICS COMPETITION

(A Mathematics Competition for Undergraduate Students)

The competition is named after Madhava, who introduced in the fourteenth centurry, profound mathematical ideas that are now part of Calculus. His most famous achivements include the Madhava – Leibnitz series for π, the Madhava – Newton power series for sine and Cosine functions and a numerical value of π that is accurate to eleven decimal places. The “Madhava School”, consisting of a long chain of teachers-students continuity, flourished for well over two centuries, making significant contributions to mathematics and astronomy.

ELIGIBILITY

The competition is basically meant for the S. Y. B. Sc. / S. Y. B. Sc. (Computer Science) students with Mathematics as one of the subjects and thus the question paper will be set with the background of these students in mind. However, interested students of F. Y. B. Sc. and T. Y. B. Sc. (Mathematics) may also appear.The merit list of T. Y. B. Sc. students will be made separately.

A Mathematics Competition for S.Y.B.Sc. Students
Interested students of F.Y.B.Sc. and T.Y.B.Sc. are also eligible.
(A seperate merit list of T.Y.B.Sc. will be prepared.)

TOPICS

Calculus of one variable (sequences, series, continuity, differentiation etc.), elementary algebra (polynomials, complex numbers and integers), elementary combinatorics, general puzzle type/ mixed type problems.

PRIZES

After the analysis of the results, first few students above a certain cut-off, with a reasonable gap after that will get cash prizes. Some more students will get the consolation prizes. All the students appearing for the competition will receive a participation certificate.

A registration fee of Rs. 50/- will be charged to the students for ensuring his/her seriousness in the enrollment and assurance of participation. The colleges will send a list of candidates from their college in the prescribed format along with the registration fees to the center coordinator. A receipt in the name of Principal of that college will be given.

The last date for the registration is November 15, 2014.

There will be no provision for revaluation of the answer sheets.