# What Does The Notation of Union Over K represent?

I am currently studying about polynomial time in Comp Theory. The definition for polynomial time given in my textbook as follows: $$P= \bigcup_{k}TIME(n^k)$$ I have not seen the Union over k notation before. K is a constant. A quick introduction to the meaning of this notation would be enormously helpful.

Thank you

• Simple: $k$ is not a constant here. It's a bound variable. – Raphael Apr 10 '16 at 12:28
• This is a good question! That notation is indeed confusing, especially if this is the first time you see it. As the other comments/answers say, it is a shorthand for $\cup_{k=1}^\infty$. – Ran G. Apr 11 '16 at 0:55

Authors often leave out the domain of the union if it is clear within the context. Here, $k$ is a natural number, i.e. $P = \bigcup\limits_{k \in \mathbb{N}} TIME(n^k)$. This is what it means for a complexity function to be bounded by a polynomial: there is a constant $k$ such that the function is in $\mathcal{O}(n^k)$.
• @PenguinsAndApples No, not the domain; that's $\mathbb{N}$. It's a bound variable. – Raphael Apr 10 '16 at 12:28