Other sources may define \(\mathbb{N}\) differently, so check each source’s definition before reading its formulas:
When a formula’s meaning depends on whether \(0\) is included, we write the set out explicitly, for example \(\mathopen{}\left\{0, 1, 2, \ldots\right\}\mathclose{}\) or \(\mathopen{}\left\{1, \ldots, n\right\}\mathclose{}\).
Source: Wikipedia, “Natural number”, “Terminology and notation” and “Zero as natural number”, which cites ISO 80000-2:2019 and the textbooks using each notation.
Example 10 (Floors)
The graph of \(\mathopen{}\left\lfloor x\right\rfloor\mathclose{}\) is a staircase: each step starts at an integer, with a filled dot, and stops before the next integer, at an open dot.
# one step per integer k, from x = k to x = k + 1 at height k
k <- -3:2
steps <- tibble::tibble(x = k, xend = k + 1, y = k)
floor_plot <- ggplot2::ggplot(steps) +
ggplot2::geom_segment(ggplot2::aes(x = x, xend = xend, y = y, yend = y)) +
ggplot2::geom_point(ggplot2::aes(x = x, y = y), size = 2) +
ggplot2::geom_point(
ggplot2::aes(x = xend, y = y),
size = 2, shape = 21, fill = "white"
) +
ggplot2::labs(x = "x", y = "floor of x")
floor_plot