23666: HW 5

Problem 21. 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 22. 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 23. 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 24. Let $R$ be a relation on $A$.

1. Show that $R$ is reflexive iff $I_A\subseteq R$. Here $I_A$ denotes the identity relation on $A$: $I_A=\{(a,a) \ |\ a\in A\}$.
2. Show that $R$ is symmetric iff $R=R^{-1}$.
3. Show that $R$ is transitive iff $R\circ R\subseteq R$.

Problem 25. Let $R$ and $S$ be two equivalence relations on a non-empty set $X$. Prove or disprove:

1. $R\cap S$ is an equivalence relation.
2. $R\cup S$ is an equivalence relation.