Questions
- What does the set builder notation for $S\circ R$ look like? I'm having the most trouble knowing when there is too much information or not enough information on either side of the 'such that'. $$S\circ R=\{ a\in A\land c\in C\land\exists b\in B\mid (a,b)\in R\land (b,c)\in S\}$$ $$S\circ R=\{(a,c)\mid(\exists b\in B)\rightarrow((a,b)\in R\land(b,c)\in S)\}$$
- A relation $R_1$ is difined on the set of integers such that $$R_1=\{(a,b)\mid a>b,\:\:\:a,b\in\mathbb{Z}\}$$ How would I write the set builder notation for the recursive composite of $R^n_1$ in terms of $a,b$ and $n$?
- As a part of question #2; In order for me to understand why the set builder definition looks the way it does I need to understand how to find $R^n_1$ for the relation $R_1$. What is a mathematical process, not an intuitive guess and check solution, that I can employ to see what is happening with respect to the given relation?
Current Understandings
- Set-builder notation is a mathematical notation to describe a set in general form. It's short hand to cut down long worded explanations. A set is usually expressed using curly brackets : $\{\cdots\}$. The set builder notation is formulated by {(a variable)("such that")(logical predicate)}
ex: The set containing all positive real numbers:
Denoted in set builder notation as: $$\{x\mid x\in\mathbb{R}\land x>0\}$$
- A binary relation on a set $A$ (of $n$ elements) is a group including pairs of elements from the set $A$. Namely, a relation $R$ on a set $A$ is a subset of $A\times A$ containing $2^{n^2}$ relations.
ex: The relation "less than" is the set containing pairs of numbers where the first number is less than the second number.
Denoted in set builder notation as: $$R=\{(a,b)\mid a<b\}$$
- When you take the composition of multiple relations you join the set mapping of one relation to another (and so on). Let $R$ be a relation from set $A$ to set $B$ and $S$ a relation from $B$ to a set $C$. $$R\subseteq A\times B\text{ and }S\subseteq B\times C$$ The composite of $R$ and $S$ (denoted by $S\circ R$) is the relation consisting of ordered pairs $(a,c)$, where $a\in A,c\in C$, and for which there exists an element $b\in B$ such that $(a,b)\in R$ and $(b,c)\in S$.
(QUESTION 1)
ex: $R$ is a relation on two sets such that $R=\{(1,1),(1,4),(2,3),(3,1),(3,4)\}$ and $S$ is a relation on two sets such that $S\{(1,0),(2,0),(3,1),(3,2),(4,1)\}$
$$S\circ R=\{(1,0),(1,1),(2,1),(2,2),(3,0),(3,1)\}$$
- Continue to let $R$ be a relation on a set $A$. When composing the parent relation with itself, the powers $R^n, n=1,2,3,\dotsc,$ are defined recessively by $R^1=R$ and $R^{n+1}=R^n\circ R$. This shows that $R^2=R\circ R, R^3=R^2\circ R, and so on.$
(QUESTION 2,3)