CS 381 Solutions to Homework 10

pp. 382 - 383

4 a) Not reflexive, not symmetric, antisymmetric, and transitive.
b) Reflexive, symmetric, not antisymmetric, and transitive.
c) Reflexive, symmetric, not antisymmetric, and transitive.
d) Reflexive, symmetric, not antisymmetric, and not transitive.

18 a) $R_{2}$, b) $R_{1}$, c) $\emptyset$,
d) $\{(1,1), (2,1), (2,2), (3,1), (3,2), (3,3)\}$

28 a) Since $R$ and $S$ are reflexive, for an arbitrary $x \in A$, $(x,x) \in R$ (and $(x,x) \in S$).
Hence $(x,x) \in R \cup S$. Hence $R \cup S$ is reflexive.
b) Since $R$ and $S$ are reflexive, for an arbitrary $x \in A$, $(x,x) \in R$ and $(x,x) \in S$.
Hence for an arbitrary $x \in A$, $(x,x) \in R \cap S$. Hence $R \cap S$ is reflexive.

pp. 396

12. Omitted

17.

#	Reflexive	Irreflexive	Symmetric	Antisymmetric	Transitive
13	No	Yes	No	No	No
14	Yes	No	No	No	No
15	No	No	Yes	No	No

pp. 406 - 407

2. R is symmetric. Hence it is its own symmetric closure.

5. Add a loop at each vertex to the graph.

9. Symmetric closure of 5: $\{(a,b), (b,a), (a,c), (c,a), (b,d), (d,b), (c,d), (d,c)\}$

20 b) $(a, b)$ in $R^{3}$ means that there is a two-stop (three leg) airline flight from city $a$ to city $b$.

24. Not necessarily. For example let $R = \{ < a, b >, < b, a > \}$ on the set $\{a, b\}$. Then $R^{2} = \{ < a, a >, < b, b > \}$, which is not irreflexive.