Question

Show that if A and B are sets with the same cardinality, then |A|<=|B| and |B|<=|A|

Answer 1

Given that , $A$ and $B$ are two set with same cardinality .i,e, $|A|=|B|$

Claim: $|A|\leq |B|$ and $|B|\leq |A|$

Since, cardinality of $A$ and $B$ are equal .Therefore there exist a bijective function $f:A\rightarrow B$

Since $f$ is one-one and $f(A)\sube B$

$\therefore |A| \leq |B|$

Again , $f^{-1}:B\rightarrow A$ is a bijective map .

Therefore $f^{-1}$ is one-one and $f^{-1}(B)\sube A$

Hence ,

$|B|\leq |A|$