Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

On wikipedia, it is stated that the set

$$f(x)=\{q\in\mathbb{Q}|q\leq x\}$$ is an inclusion map. If I define $a,b$ $\in\mathbb{R}$ such that $a\ne b$ and don't see how $\{q\in\mathbb{Q}|q\leq a\}\ne \{q\in\mathbb{Q}|q\leq b\}$ since bot sets should be infinite.. What am I missing here?

share|cite|improve this question
Two sets are equal iff they contain the same elements, not if they have the same cardinality. – Thomas Feb 24 '14 at 16:00

The two sets are both infinite, but that does not imply that they are equal. For example, $\mathbb{N}$ and $\mathbb{Z}$ are both countably infinte but of course not equal.

Assuming w.l.o.g. $a<b$, then there exists some element $c \in \mathbb{Q}$ such that $a<c<b$ where $$c \not\in \{q \in \mathbb{Q} | q \leq a\}$$ but $$c \in \{q \in \mathbb{Q} | q \leq b\}$$

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.