CRN 12107: HW 5
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'''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 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 $ | + | '''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. | '''Problem 25.''' Exercise 3.2 #13. | ||
Latest revision as of 12:29, 23 October 2013
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 $C$.
- Prove or disprove: Dom$(S\circ R)\subseteq$ Dom$(R)$.
- 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.
