# SI Worksheet 10/10/21

1. Consider the following graphs to be representations of ARSs. For each of the graphs, determine if its ARS is Terminating, Confluent, and has Unique Normal Forms. Then rewrite the ARS in the standard notation (i.e. A={...} R={...})
1. 
2. 
3. 

2. Consider the following ARSs. Rewrite each ARS as a graph then determine the properties of the ARS as in question 1.
1. $A = \{a, b, c, d, e, f\}$
$R = \{(a,b),(a,c),(c,e),(e,a),(f,d)\}$

2. $A = \{a, b, c, d\}$
$R = \{(a,a),(a,b),(a,c),(b,a),(b,b)(b,d),(d,d)\}$
3. Looking at the graphs, what generalizations can be made about each of the three properties? In other words, what do the graphs reveal about the nature of each property?
4. Recall relations from Discrete Math as well as the ideas of Reflexitivity, Symmerty, and Transitivity. The [discrete math review](https://hackmd.io/@alexhkurz/SJ1cc-dDr) that Dr. Kurz wrote is a good reminder on these ideas. ARSs can be considered relations themselves: the relation $aRb$ implies $(a,b)\in R$
For each of the following ARSs, if it is a graph, write it in standard notation, and if it's in standard notation draw its graph. Then determine if the relation it represents is reflexive, symmertic, or transitive.

1. $A=\{a,b,c,d\}$
$R=\{(a,b),(b,a),(c,d),(d,c)\}$

2. 

3. $A=\{a,b,c,d\}$
$R=\{(a,a),(b,b),(c,c),(d,d)\}$

4. 

6. Recall the idea of Transitive Closure. In simple terms, it is just a version of a relation that allows "skipping steps." Which of the relations from question 5 is the transitive closure of another? What is the transitive closure of the following relations: Parent, Child, Successor (in terms of successor notation) What about the ARSs in question 1?

*If you'd like to play around with the same graph creation tool I used, [here's a link](https://csacademy.com/app/graph_editor/)*