# Set-Builder Notation Where Numbers Increase With Fixed Value

I'm a little shaky with set-builder notation and I was wondering how to define the following range mathematically: $$[0.05, 0.075, 0.1, 0.125, 0.15]$$

I understand that, for instance, $$\{x \in \mathbb{R} : 0.05 \leq x \leq 0.15\}$$ specifies that the variable $$x$$ is any real number between $$0.05$$ and $$0.015$$, but how would I write, in set-builder notation, that the variable $$x$$ is limited by $$0.05$$ and $$0.15$$ but only includes those numbers at intervals of $$0.025$$?

In the proposed notation, I want to make clear that the value $$x$$ lies between the range of $$0.05$$ and $$0.15$$ and that the values that belong in $$x$$ are only those values which are increments of $$0.025$$.

• But there are only five numbers! List them. Commented Oct 8, 2021 at 16:32
• What about $\{ 0.025 n | n \in \mathbb Z , 2 \le n \le 6 \}$? Commented Oct 8, 2021 at 16:37
• @MauroALLEGRANZA I'm not entirely sure what you mean, this is the range that I want to encapsulate. Beginning at 0.05 and increasing at 0.025 increments until the limit of 0.15. Commented Oct 8, 2021 at 16:39
• @MohsenShahriari Is it at all possible to reformulate this by making the limits clearer? I'm not sure this is what I want to convey. Commented Oct 8, 2021 at 16:44
• By "limits" do you mean $0.05$ (the least number in the list) and $0.15$ (the greatest number in the list)? Or something else? And if there is something specific that you want to convey, it's better that you edit your post and add some explanation. As it stands, the suggestions by @MauroALLEGRANZA and myself seem to be perfect suggestions. Commented Oct 8, 2021 at 16:50

• Write $$[0.05, 0.15] \cap 0.025 \mathbb Z$$. (The intersection of $$[0.05,0.15]$$, the interval of all real numbers in this range, with $$0.025\mathbb Z$$, the set of all integer multiples of $$0.025$$.)
• Write $$\{x \in \mathbb R : 0.05 \le x \le 0.15 \text{ and } \frac{x}{0.025} \in \mathbb Z\}$$. (The restriction that $$\frac{x}{0.025}$$ is an integer also implies that $$x$$ is a multiple of $$0.025$$. We could simplify this to $$40x \in \mathbb Z$$, but then it's not immediately clear that the step size is $$0.025$$.)
• Write $$\{0.05, 0.075, 0.1, 0.125, 0.15\}$$. (This is the best way when the set is not too large. But you wouldn't want to do this if we were going from $$1$$ to $$100$$ in increments of $$0.025$$.)
• Write a description in words like "the set of numbers starting at $$0.05$$ and going up by $$0.025$$ until reaching $$0.15$$". (The first goal of mathematical notation is being understood. Often the best way to do this is with words, not symbols.)
The first two methods rely on the fact that the starting point $$0.05$$ is a multiple of $$0.025$$. If we wanted to go from $$0.05$$ to $$0.15$$ in multiples of $$0.02$$, we'd need a more careful restriction, such as $$\{x \in \mathbb R: 0.05 \le x \le 0.15 \text{ and } \tfrac{x-0.05}{0.02} \in \mathbb Z\}.$$
• @AlexanderHepburn Yet (as is mentioned in the last paragraph of the post) it has the disadvantage that you need to verify $0.05 \in 0.025 \mathbb Z$ and $0.15 \in 0.025 \mathbb Z$ to see that those are indeed the "limits". For example, $[ 0.04 , 0.16 ] \cap 0.025 \mathbb Z$ indicates the same set, while $0.04$ and $0.16$ are virtually irrelevant. Commented Oct 8, 2021 at 18:13