23

Reading a book on algorithms. Can someone explain the meaning of the mathematical symbol ∃?

Mike Dinescu
  • 54,171
  • 16
  • 118
  • 151
Yada
  • 30,349
  • 24
  • 103
  • 144
  • Discrete structures during the winter session? Ouch. – John Bellone Dec 23 '09 at 15:00
  • 5
    Belongs on MathOverflow http://mathoverflow.net/ – Jon Seigel Dec 23 '09 at 15:02
  • 3
    Set theory and predicate logic are 100% programming related, whomever voted to close. Although MathOverflow.net is better for pure math questions, obviously. – Randolpho Dec 23 '09 at 15:07
  • Given that this kind of predicated logic is a staple of CS courses, I don't see why it should be closed. – Pod Dec 23 '09 at 15:09
  • @Randolpho: Definitely. However, that has no connection to the question that was asked. – Jon Seigel Dec 23 '09 at 15:10
  • 25
    MO is not a math equivalent of SO. This question would be closed there. ("intended audience is professional mathematicians, mathematics graduate students, and advanced undergraduates." from FAQ) – sdcvvc Dec 23 '09 at 15:10
  • 8
    I'm voting to reopen. A similar question was asked recently about ∀ (see http://stackoverflow.com/questions/1925979/unfamiliar-symbol-in-algorithm-what-does-mean). Also it definitely does NOT belong on MathOverflow -- that's for Serious mathematical research questions; a question like this would get shut down before you could say ∀ B Ↄ.... – Jason S Dec 23 '09 at 15:28
  • 1
    Agreed this should be allowed. Developers should know how to write invariants, pre-conditions and post-conditions. Perfectly reasonable for someone learning the trade to ask this question. – Chris McCauley Dec 23 '09 at 15:36
  • http://meta.stackexchange.com/questions/9658/i-was-told-thered-be-no-math – Jon Seigel Dec 23 '09 at 15:40
  • @Jon Seigel: The top answer, from your link - "I would say Math questions are suitable as long as they tie into a programming concept." The explanation is required to understand an algorithm text. Voting to reopen. – ire_and_curses Dec 23 '09 at 17:01
  • @ire_and_curses: The question here does not reference any algorithm or programming concept, therefore it is a math question. – Jon Seigel Dec 23 '09 at 17:21
  • 3
    Someone who does not understand what ∃ means might not know that this is a math question. Should questions on O notation be closed because they are math questions? While it's a mathematical notation, it's usually used in analysis of algorithms. How is someone reading an algorithms text supposed to know that ∃ is any different? – Brian Campbell Dec 23 '09 at 22:19
  • Questions on O notation are always linked to algorithms, or a specific example of an algorithm. What is the context in this question? Fair point about not knowing this is a math question. However, it makes me wonder why the book the OP is reading does not contain a definition for the terminology? Any textbook I've read has a dictionary of symbols somewhere in it. – Jon Seigel Dec 24 '09 at 13:31
  • @Jon Seigel: The book almost surely does, the OP just overlooked it. Even the most advanced textbooks often have a table of symbols page somewhere just because there are inconsistencies between authors. – jason Dec 24 '09 at 14:37
  • This is simply inane. Are we going to have a question for every mathematical symbol on Earth? Let me get it going: What is א0? – Sinan Ünür Dec 25 '09 at 02:19
  • This belongs on math.stackexchange.com, which didn't exist during all of the previous discussion. – Eric Wilson Sep 12 '11 at 15:26
  • Three months later, I'm curious why this has not been migrated to Math.SE. – Eric Wilson Dec 21 '11 at 16:25

5 Answers5

55

It is called a quantifier. It means "there exists".

When used in an expression such as 

∃x s.t. x > 0

It means "There exists a number x such that x is greater than 0."

Its counterpart is ∀, which means "for all". It's used like this:

∀x, x > 0

Which means "For any number x, it is greater than 0."

Welbog
  • 59,154
  • 9
  • 110
  • 123
11

It is the "existential quantifier" as opposed to the upside-down A (∀) which means "universal quantifier." It should be read as "there exists" or "for some". It is a predication that means that some relation or property holds true for at least one object in the domain.

Examples:

An integer n is composite if ∃ integer m such that m > 1 and m < n with n divisible by m.

An integer n is prime if ∀ integer m such that m > 1 and m < n it is true that n is not divisible by m.

A function f is continuous on a metric space (X, d) if ∀x∀ε>0∃δ>0 | ∀y d(x, y) < δ => d(f(x), f(y)) < ε

jason
  • 236,483
  • 35
  • 423
  • 525
  • Oh no not the epsilons and deltas! Calculus 1 is flooding back to me now. I have only you to blame, Jason. – Welbog Dec 23 '09 at 15:17
  • 1
    Ah, whom am I kidding? I loved Calculus 1. That's why I took Calculus 2 and Calculus 3! Thanks for that trip down memory lane. – Welbog Dec 23 '09 at 15:18
  • Alternating between quantifiers produces formulas which are both hard to understand and hard to handle algorithmically. For example, the definition of continuity has the pattern ∀∃∀ (the ∀x∀y is missing in the example). – starblue Dec 24 '09 at 08:05
2

More Info on Predicate Logic

Matt
  • 3,664
  • 3
  • 33
  • 39
2

It is called existential quantifier and being followed by x, it means there exists at least one x

psihodelia
  • 29,566
  • 35
  • 108
  • 157
1

For future reference, wikipedia has a table of mathematical symbols, with an explanation of the meaning(s) of each one.

Dave Kirby
  • 25,806
  • 5
  • 67
  • 84