Unfamiliar symbol in algorithm: what does ∀ mean?

Question

I'm reading about an algorithm (it's a path-finding algorithm based on A*), and it contains a mathematical symbol I'm unfamiliar with: ∀

Here is the context:

v(s) ≥ g(s) = min_s'∈pred(s)(v(s') + c(s', s)) ∀s ≠ s_start

Can someone explain the meaning of ∀?

Solution 1

That's the "forall" (for all) symbol, as seen in Wikipedia's table of mathematical symbols or the Unicode forall character (\u2200, ∀).

Solution 2

The upside-down A symbol is the universal quantifier from predicate logic. (Also see the more complete discussion of the first-order predicate calculus.) As others noted, it means that the stated assertions holds "for all instances" of the given variable (here, s). You'll soon run into its sibling, the backwards capital E, which is the existential quantifier, meaning "there exists at least one" of the given variable conforming to the related assertion.

If you're interested in logic, you might enjoy the book Logic and Databases: The Roots of Relational Theory by C.J. Date. There are several chapters covering these quantifiers and their logical implications. You don't have to be working with databases to benefit from this book's coverage of logic.

Solution 3

In math, ∀ means FOR ALL.

Unicode character (\u2200, ∀).

Solution 4

Can be read, "For all s such that s does not equal s[start]"

Solution 5

yes, these are the well-known quantifiers used in math. Another example is ∃ which reads as "exists".

http://en.wikipedia.org/wiki/Quantification

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Comments

[Admin]

More comments are getting flagged, and if they're not pertinent to the question, they are deleted.

[devuxer]

That would make sense: "...for all s unequal to s[start]"

[user1066101]

They call it the "Universal Qualifier". Compare with the "Existential Qualifier". en.wikipedia.org/wiki/Universal_quantification

[R. Martinho Fernandes]

@S.Lott: nitpick s/qualifier/quantifier

[devuxer]

Thanks and +1 for the link to the table of symbols. I will use that next time I'm stumped (searching Google for ∀ turned up no records).

[R. Martinho Fernandes]

lol, I had never thought about googling for symbols. And apparently I didn't lose anything.

[starblue]

+1 for mentioning ∃ (U+2203 THERE EXISTS). Actually ∀ and ∃ are used in predicate calculus in general, be it first-order or higher-order. For a second-order example, in the induction axiom of Peano arithmetic you quantify over predicates and write ∀P.

[subrat71]

Thanks for pointing that out. I revised the reference per your suggestion.