Suppose $A$ is a finite set of cardinality $n$. And Let $B$ be a subset of $A$ and the cardinality of $B$ equals $n$. Then $B=A$.
Many texts use this fact very frequently but it seems that they just take it for granted. How can I prove this rigorously? Any help will be appreciated.
Best Answer
Since $B \subseteq A$, we can partition $A = B \cup (A \setminus B)$. These sets are disjoint. Taking cardinalities, we see $n = n + \left|A \setminus B\right|$, which implies $|A \setminus B| = 0$, hence $A \setminus B = \emptyset$, so $A = B$.