Prove: If |A| = n and A is equinumerous to B, then |B| = n
Prove: Let A and B be finite sets. Conjecture a formula for |A U B| in the case that A n B doesn't equal the empty set
If $\displaystyle |A|=n$ it means there is a bijection $\displaystyle \theta : A\mapsto n$, and $\displaystyle |A|=|B|$ means there is a bijection $\displaystyle \eta: A\mapsto B$. Thus, the mapping $\displaystyle \theta \circ \eta^{-1}$ will be a bijection from $\displaystyle B$ to $\displaystyle n$.
If $\displaystyle |A|=n$ and $\displaystyle |B|=m$ then $\displaystyle |A\cup B| \leq n+m$ and we have equality when $\displaystyle A\cap B=\emptyset$.Prove: Let A and B be finite sets. Conjecture a formula for |A U B| in the case that A n B doesn't equal the empty set