In an earlier post, I discussed the basic and most important aspects of Set theory, Functions and Real Number System. In the same, there was a significant discussion about the union and intersection of sets.

Restating the facts again, given a collection $ \mathcal{A}$ of sets, the union of the elements of $ \mathcal{A}$ is defined by

$ \displaystyle{\bigcup_{A \in \mathcal{A}}} A := {x : x \in A \textrm{ for at least one } A \in \mathcal{A} }$ .

The intersection of the elements of $ \mathcal{A}$ is defined by

$ \displaystyle{\bigcap_{A \in \mathcal{A}}} A := {x : x \in A \textrm{ for every } A \in \mathcal{A} }$ .

But what if the collection is empty? i.e., what if the collection contains no sets?

From the first day at Topology class in Gorakhpur University , it was the very first confusion which troubled many of classmates. Topology’s Professor was elaborating the finite intersection of sets in reference to topological space — and everyone in the classroom was enjoying it. To end of the lecture, as he wrote,

the intersection of members of empty collection equals to the universal set.

Which mathematically, Professor wrote $ \bigcap \emptyset = X$ where $ X$ is universal set. The quote was okay and significant but the mathematical notion of the same was confusing, as he represented an empty collection by “empty-set notation $ \emptyset$ ” . Professor easily asserted this to be an agreement among some mathematicians and even though the result is useful in some problems, he skipped a proper proof of it. [That’s really bad, Sir!]  At first instant, this problem lead me to a single result: $ \bigcap {\emptyset} = \emptyset$ , if $ \emptyset$ was usual null-set. But in the case, where $ \emptyset$ was an empty collection – I couldn’t reach to any result.

There came Munkres’s Topology book to the rescue. It’s second edition has a little note about the same on Chapter 1 page 12 .  He writes:

There is no problem with these definitions if one of the elements of $ \mathcal{A}$ happens to be the empty set. [In each such case of arbitrary union  and intersection the results is $ \emptyset$ .] But it is tricky to decide what (if anything) these definitions mean if we allow $ \mathcal{A}$ to be the empty collection. Applying the definitions literally, we see that  no element $ x$ satisfies the defining property for the union of the elements of $ \mathcal{A}$ . So it is reasonable to say that

$ \displaystyle{\bigcup_{A \in \mathcal{A}}} A := \emptyset$ if $ \mathcal{A}$ is empty.

On the other hand , every $ x$ satisfies (vacuously) the defining property for the intersection of the elements of $ \mathcal{A}$ . The question is, every $ x$ in what set? If one has a given large set $ X$ that is specified at the outset of the discussion to be one’s “universe of discourse”, and one considers only subsets of X throughout, it is reasonable to let

$ \displaystyle{\bigcap_{A \in \mathcal{A}}} A := X$ when $ \mathcal{A}$ is empty.

Not all Mathematicians follow this convention, however. …

 

So, according to Munkres (with some very good arguments), the intersection of elements of an empty collection of subsets of a set (say) $ E \subseteq X$ equals to the universal set $ X$ . In other words, nothing is everything. There must be any theoretical or logical proof of it! There must be.

Leave a Reply

Your email address will not be published. Required fields are marked *

This site uses Akismet to reduce spam. Learn how your comment data is processed.

You May Also Like

Mathematical Logic – The basic introduction

What is Logic? If mathematics is regarded as a language, then logic is its grammar. In other words, logical precision has the same importance in mathematics as grammatical accuracy in a language. As linguistic grammar has sentences, statements— logic has them too. After we discuss about Sentence & Statements, we will proceed to further logical theories . Sentences & Statements…

Three Children, Two Friends and One Mathematical Puzzle

Two close friends, Robert and Thomas, met again after a gap of several years. Robert Said: I am now married and have three children. Thomas Said: That’s great! How old they are? Robert: Thomas! Guess it yourself with some clues provided by me. The product of the ages of my children is 36. Thomas: Hmm… Not so helpful clue. Can…

Largest Prime Numbers

What is a Prime Number? An integer, say $ p $ , [ $ \ne {0} $ & $ \ne { \pm{1}} $ ] is said to be a prime integer iff its only factors (or divisors) are $ \pm{1} $ & $ \pm{p} $ . As? Few easy examples are: $ \pm{2}, \pm{3}, \pm{5}, \pm{7}, \pm{11}, \pm{13} $ …….etc.…

Milnor wins 2011 Abel Prize

The Norwegian Academy of Science and Letters has decided to award the Abel Prize for 2011 to John Milnor, Institute for Mathematical Sciences, Stony Brook University, New York “for pioneering discoveries in topology, geometry and algebra”. The President of the Norwegian Academy of Science and Letters, Øyvind Østerud, announced the winner of this year’s Abel Prize at the Academy in…

11-12-13 – Last such sequence of the century

11th December 2013 (or in short 11-12-13) is just a few hours away here. It’s the last date of the 21st century with  such an extraordinary pattern of numbers.  In any calendar of the world, such date will be seen in 22nd century after 87 years and 54 days from today — on February 3, 2101, if all kind of date…

Chess Problems

In how many ways can two queens, two rooks, one white bishop, one black bishop, and a knight be placed on a standard $ 8 \times 8$ chessboard so that every position on the board is under attack by at least one piece? Note: The color of a bishop refers to the color of the square on which it sits,…