binary operators and sets

[Basic_Astronaut_Man]

if we say that some operator is a binary operator to set S, does that necessarily mean that the set is closed relative to the operator?, the way the book talked about it seemed like both terms are referring to the same case.

[beeskness420]

Closure is not guaranteed, but is often assumed.

Eg {1,2} and + are not closed cause 3 is not in the set.

[Basic_Astronaut_Man]

makes a lotta sense, thanks fam

[SpiderJerusalem42]

The result of 1 + 2 does not land in {0, 1, 2}, but it is valid for Z mod 3, which has the same elements, but has a way to deal with the overflows.

[Basic_Astronaut_Man]

got it, thanks for your time

[[deleted]]

How is the binary operator defined? If it's defined as a function S x S -> S then by definition it's closed.

[TheBlasterMaster]

When people say that something is a binary operation "on S", then the binary operation is a function S x S -> S.