In the following problem, what does the "circle" between set names represent? What exactly is this problem asking me to do?
Consider the following relations on Z:
$R1 = \{(x, y) | y = x + 1\}$
$R2 = \{(x, y) | y = x − 1\}$
$R3 = \{(x, y) | y = 2x + 1\}$
$R4 = \{(x, y) | y = 2x − 1\}$
Describe each of the following composite relations in set builder notation.
R1 ◦ R1
R2 ◦ R1
R3 ◦ R1
R4 ◦ R1
R1 ◦ R2
R2 ◦ R2
R3 ◦ R2
R4 ◦ R2
R1 ◦ R3
R2 ◦ R3
R3 ◦ R3
R4 ◦ R3
R1 ◦ R4
R2 ◦ R4
R4 ◦ R4
On
The little circle ($\circ$) is the composition operator. It operates on two relations to produce a third relation, their composition. I guess this is either homework or self-study, in which case I would check your textbook for the definition of the composition of two relations. However, Fejer and Simovici in Mathematical Foundations of Computer Science define composition in this way:
$\rho \circ \sigma = \{(x,z) \mid \text{for some }y, \: (x,y) \in \rho, \text{ and } (y,z) \in \sigma \}$
(although they call it a product and use $\bullet$, noting that the above notation and terminology is also used).
Additionally, here is a link to the wikipedia article on composition of relations.
It represents composition of the relations. In the problem with $R_a \circ R_b$ you are being asked to find the pairs $(x,y)$ such that there is some $z$ with $xR_az$ and $z R_b y$. These relations all happen to be bijections-for a given $x$ there is exactly one $y$ such that $xRy$. So if we want to find pairs $(x,y)$ belonging to $R_2 \circ R_3$ we must have $z=x-1$ and $y=2z+1=2(x-1)+1=2x-1$. Our answer would be $\{(x,y)\mid y=2x-1\}$ Can you do the rest?