how to formally define a complementary set

Question

I have a set whose elements are integer solution of the equation $f(k)=2k+5$, where $k$ is an integer from 0,1,2,3,...,
how would you formulate the complement of this set? e.g. suppose we are interested in integer values which are not of the form $2k+5$,
how would I define formally the complementary set?

Answer 1

The set you have defined is simply the odd integers, starting with $5$. Note that every odd integer can be written as $2k+5$ for some integral $k$. The restriction to non-negative $k$ gives us $5,7,9,11,13,\ldots$

Integers not in that set are either even, less than $5$, or both. You could write it as $\{x\in\mathbb{Z}|x \text{ is even, or }x<5\}$, for example.

Answer 2

Your set, which I'll call $S$, is $S = \{5, 7, 9, ... \}$. Simply put, it is any odd number greater than 3. We can formally define it as $S = \{2k+5\, |\, k \in \mathbb{N}\}$. But we can also define it as $S = \{2k+1\,|\,k\in \mathbb{Z} \,\land\,k \ge2 \}$. This means that the number is of the form $2k+1$ where $k$ is an integer and at least 2.

The complement, within the set of integers, will be any even number or any odd number not greater than 3. One way can write this is to say that $\mathbb{Z} \,\backslash\, S = \{2k\,|\,k \in \mathbb{N}\} \;\cup\; \{ k \in \mathbb{Z}\,| \,k \le 3\}$. We could also say it is $\{ x \in \mathbb{Z}\,|\, x \equiv 0 \pmod n \;\lor\; x < 5\}$.