I'm confused about these 3 selected problems. I have the solutions for each, if necessary, but I'm much more interested in understanding the material. If anyone can offer a clear, concise, and intuitive explanation for each- or even just one- I'd be entirely grateful.
Let $R$ be the relation on the set of people consisting of $(a,b)$ where $a$ is the parent of $b$. Let $S$ be the relation on the set of people consisting of $(a,b)$ where $a$ and $b$ are siblings. What are $S\circ R$ and $R\circ S$?
A) $(a,b)$ where $a$ is a parent of $b$ and $b$ has a sibling; $(a,b)$ where $a$ is the aunt or uncle of $b$.
B) $(a,b)$ where $a$ is the parent of $b$ and $a$ has a sibling; $(a,b)$ where $a$ is the aunt or uncle of $b$.
C) $(a,b)$ where $a$ is the sibling of $b$'s parents; $(a,b)$ where $a$ is $b$'s niece or nephew.
D) $(a,b)$ where $a$ is the parent of $b$; $(a,b)$ where $a$ is the aunt or uncle of $b$.
On the set of all integers, let $(x,y) \in R$ iff $xy \geq 1$. Is relation R reflexive, symmetric, antisymmetric, transitive?
A) Yes, No, No, Yes
B) No, Yes, No, Yes
C) No, No, No, Yes
D) No, Yes, Yes, Yes
E) No, No, Yes, No
At what smallest power of R do we find the connectivity relation R* in this diagram?
A) $R^* = R$
B) $R^* = \bigcup_{k=1}^2 R^k$
C) $R^* = \bigcup_{k=1}^3 R^k$
D) $R^* = \bigcup_{k=1}^4 R^k$
E) $R^* = \bigcup_{k=1}^5 R^k$
EDIT: I understand the first two pretty well. That is, I know what they're asking for, but my path to the solution doesn't seem to be correct. For the first one, what do I need to know about the way relational compositions work that will make this problem more manageable? For the second, I need help with the symmetric and antisymmetric part (how does it work here)? And the third, which I'm having the most trouble with- what exactly is $R^*$, and how does it relate to the notation in each of the answers ($\bigcup$)?