21495: HW 5
For a natural number $n$, let ${\cal D}_n$ denote the set of the divisors of $n$. For example, ${\cal D}_{42}=\{1,2,3,6,7,14,21,42\}$ and ${\cal D}_{12}=\{1,2,3,4,6,12\}$. For $m,n\in\mathbb{N}$ let $m\sqcap n$ denote the greatest common divisor of $n$ and $m$, and $m\sqcup n$ their least common multiple. For instance $6\sqcap 4=2$ and $6\sqcup 4=12$. It turns out that ${\cal D}_{42}$ with these two operations $\sqcap$ and $\sqcup$ forms a Boolean Algebra, while ${\cal D}_{12}$ does not.
Problem 21.
- Verify Boolean Algebra Law 7 for ${\cal D}_{42}$.
- Show that ${\cal D}_{12}$ does not form a Boolean Algebra.
- Conjecture for which values of $n$ the set ${\cal D}_{n}$ forms a Boolean Algebra.
Problem 22. Let $R$ and $S$ be two relations on $\mathbb{R}$: $R=\{(x,y)\in\mathbb{R}\times\mathbb{R}\ |\ y<x^2\}$ and $S=\{(x,y)\in\mathbb{R}\times\mathbb{R}\ |\ y=2x-1\}$. Find $S\circ R$ and $R\circ S$.
Problem 23. Let $R$ be a relation from $A$ to $B$. For an element $b\in B$ define the set $R_b:=\{a\in A\ |\ (a,b)\in R\}$. Show \[\bigcup_{b\in B} R_b=\mbox{Dom}\, R.\]
Problem 24. Define a relation $S$ on $\mathbb{R}$ as follows: $a\,S\,b$ if $a-b$ is irrational. Prove or disprove: $S$ is (a) reflexive, (b) symmetric, (c) transitive.
Problem 25. Let $R$ be a relation on $A$.
- Show that $R$ is reflexive iff $I_A\subseteq R$. Here $I_A$ denotes the identity relation on $A$: $I_A=\{(a,a) \ |\ a\in A\}$.
- Show that $R$ is symmetric iff $R=R^{-1}$.
- Show that $R$ is transitive iff $R\circ R\subseteq R$.
