# Dedekind cuts and cardinailty

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?

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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 $$c \in \{q \in \mathbb{Q} | q \leq b\}$$

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