Cardinalities of the intervals $(-1,1)$ and $[-1,1]$

Question

Suppose we have the set $A=(-1,1)$ and the set $B=[-1,1]$.
Prove or disprove that $|A|=|B|$.

Can anyone help give a formal proof to this? I've been stuck on it for a while.

It’s probably easiest to use the Schröder–Bernstein theorem (if you’re allowed to). — Eike Schulte, Dec 18 '19 at 08:50

score 0 · Answer 1 · answered Dec 18 '19 at 08:18

0

$A$ and $B$ both have Lebesgue measure $1$, and necessarily have the same cardinality. If you are not comfortable with the notion of Lebesgue measure, then you need to exibit a bijection between $A$ and $B$.

answered Dec 18 '19 at 08:18

Math1000

36,983

score 0 · Accepted Answer · answered Dec 18 '19 at 08:59

0

Use @EikeSchuite's suggestion: we can inject $A$ into $B$ with $x\mapsto x$, & $B$ into $A$ with $x\mapsto x/2$.

answered Dec 18 '19 at 08:59

J.G.

115,835

Cardinalities of the intervals $(-1,1)$ and $[-1,1]$

2 Answers2