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Empty Product

An empty product is a product taken over an empty set of factors. Its value is defined to be the identity element for multiplication, conventionally 1. For example,

product_(k=1)^0a_k=1.

This convention makes the recurrence equation

product_(k=1)^na_k=(product_(k=1)^(n-1)a_k)a_n

valid starting at n=1.

The logical analog is the empty conjunction, whose value is true, the identity element for AND. This agrees with vacuous truth: an assertion about every element of an empty set is true because there is no counterexample.

See also

Conjunction, Empty Sum, Factor, Identity Element, Multiplication, Product, Vacuous Truth

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References

Graham, R. L.; Knuth, D. E.; and Patashnik, O. Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley, 1994.

Cite this as:

Weisstein, Eric W. "Empty Product." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/EmptyProduct.html

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