Proof Tutorial Class, Week 22

Question One §

Question §

Give an example of finite sets $A,B,X,Y$ such that $(A\times Y)\cup (B\times Y)\ne (A\cup B)\times (X\cup Y)$.

Hint: try small sets, say, subsets of $1,2,3,4$.

Solution §

…

Question Two §

Question §

For a fixed set $S$, consider $\mathcal{P}(S)$ (set of all subsets of $S$) ordered by inclusion. For $A,B\subset S$, what is $\sup \{A,B\}$ with respect to this order? What is $\inf \{A,B\}$?

Solution §

…

Question Three §

Question §

How to tie a goat so as it can eat grass exactly within a square? It is allowed to use several leashes, with several pegs. For example, one peg and leash mean that the goat eats within a circle. It is also allowed to string one rope tightly between two pegs, and to tie the goat with leash to as small ring sliding on the first rope.

Hint: use intersections of sets.

Solution §

…