Consider the relation R on {1,2,3,4,5} given by xRy $\displaystyle \longleftrightarrow$ x + y >5
How do I make the matrix for this relation?
Is the relation transitive? Why?
I suppose the matrix for the relation would be defined as:
$\displaystyle A = (a_{ij}); \ a_{ij} = \begin{cases} 1 & i+j \geq 5 \\ 0 & i+j < 5 \end{cases}$
The relation is transitive if for any $\displaystyle x,y,z \in \{1,2,3,4,5\} : x+y \geq 5 , y+z \geq 5 \Rightarrow x+z \geq 5$
But what happens if you take $\displaystyle x=1, y=5, z=2$? Is $\displaystyle x+z \geq 5$?