Symbol in Linear Algebra

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I'm newbie in linear algebra and I do not understand the symbol that is selected with blue color.

What does this symbol means?
What is the purpose to use this symbol?
What context is this symbol being used?
Why do you use this symbol?

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There are 2 best solutions below

0
On

it means

O is an element of S.
0
On

When it says $\vec{0}\in S$ what it means is "The zero or null vector in the space $S$"