CRN 12107: 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 the set $A$ to the set $B$, and $S$ be a relation from the set $B$ to the set $C$.

1. Prove or disprove: Dom$(S\circ R)\subseteq$ Dom$(R)$.
2. Prove or disprove: Rng$(S\circ R)\subseteq$ Rng$(S)$.

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. Exercise 3.2 #13.