I'm a little bit confused about why even numbers are closed under multiplication.

Closure under multiplication is $$\lambda u \ \in V $$ where $V$ is a vector space. If $\lambda = 1.5$, then $\lambda u$ is an odd number right? Therefore, it should not be closed under multiplication?

  • 3
    $\begingroup$ 1.5 isn't even. $\endgroup$ – person Oct 12 '20 at 18:34
  • $\begingroup$ Even integers are only closed under multiplication by even integers. And no need to mix it with vector spaces. $\endgroup$ – Dietrich Burde Oct 12 '20 at 18:35
  • $\begingroup$ The closure under multiplication is meant for a group or a ring, not a vector space. $\endgroup$ – Ninad Munshi Oct 12 '20 at 18:38
  • $\begingroup$ Does $\lambda$ necessarily have to be even? I thought just the result has to be even. $\endgroup$ – CSCSCSCSCSCSCSCSCS Oct 12 '20 at 18:43

A closed binary operation merely means that the elements remain in the same set, which is to say the operation is a function of the form $X \times X \rightarrow X$. So for example the natural numbers are closed under addition because when you add two naturals numbers together the answer is still a natural number.

In this case the even numbers are closed under multiplication because $2m \times 2n = 2(2mn)$ which is even.


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