CRN 12107: HW 3
Problem 11. Let $A$ be a set, and let ${\cal K}$ be a collection of sets. Show that \[A\cap(\bigcup_{B\in{\cal K}} B)=\bigcup_{B\in{\cal K}}(A\cap B).\]
Problem 12. Let $A$ be a proper subset of some set $U$, and let $x\in U\setminus A$. Let ${\cal B}$ consist of all sets of the form $C\cup\{x\}$ with $C\in{\cal P}(A)$, in other words \[{\cal B}=\{C\cup\{x\}\ |\ C\in{\cal P}(A)\}.\] Show that
- ${\cal P}(A\cup \{x\})={\cal P}(A)\cup {\cal B}$.
- ${\cal P}(A)\cap {\cal B}=\emptyset$.
Problem 13. Use the previous problem to show that ${\cal P}(A)$ has $2^n$ elements, when $A$ has $n$ elements.
Problem 14. Prove for all natural numbers $n\geq 5$: $2^n>n^2$.
Problem 15. Let $n\in\mathbb{N}$. Conjecture a formula for the expression \[a_n=\frac{1}{1\cdot 2}+\frac{1}{2\cdot 3}+\frac{1}{3\cdot 4}+\cdots +\frac{1}{n\cdot (n+1)}\] and prove your conjecture by induction.
