I am working on this problem
- Let R be the relation on the set {1, 2, 3, 4, 5} containing the ordered pairs {(1,1), (1, 2), (1,3), (2,3), (2,4), (3,1), (3,4), (3,5), (4,2), (4,5), (5,1), (5,2), (5,4)}
Find
a.$R^2$
b.$R^5$
Here is my work so far, I chose to represent R as a zero one matrix (one means in relation, zero means not in relation) Here is R, where row number is the first in the pair while column number is the second in the pair (both one based)
$\quad$1$\quad$1$\quad$1$\quad$0$\quad$0
$\quad$0$\quad$0$\quad$1$\quad$1$\quad$0
$\quad$1$\quad$0$\quad$0$\quad$1$\quad$1
$\quad$0$\quad$1$\quad$0$\quad$0$\quad$1
$\quad$1$\quad$0$\quad$0$\quad$1$\quad$0
And based off R, here's $R^2$(boolean product of R and itself)
$\quad$1$\quad$1$\quad$1$\quad$1$\quad$1
$\quad$1$\quad$1$\quad$0$\quad$1$\quad$1
$\quad$1$\quad$1$\quad$1$\quad$1$\quad$1
$\quad$1$\quad$0$\quad$1$\quad$1$\quad$0
$\quad$1$\quad$1$\quad$1$\quad$1$\quad$1
And R$^3$ is just a matrix of all ones
So my answers are
a.All possible combinations except (2,3), (4,2), (4,5), (5,4)
b.All possible combinations because for all n >= 3, $R^n = R^3$
Does everything look right? Are there any coordinates I missed in either zero one matrix?