1 votes 1 votes Let A={1,2,3,4,5,6,7} What will be no of symmetric relations on A that contains exactly 4 ordered pairs? Set Theory & Algebra relations discrete-mathematics set-theory&algebra + – Arnabi asked Feb 19, 2017 recategorized Feb 19, 2017 by Arnabi Arnabi 676 views answer comment Share Follow See all 0 reply Please log in or register to add a comment.
Best answer 7 votes 7 votes One possible relation , $\begin{align*} &R = \left \{{\color{blue}{\left ( 1,1 \right )}},{\color{red}{\left ( 1,3 \right )}},{\color{blue}{\left ( 2,2 \right )}},{\color{red}{\left ( 3,1 \right )}} \right \} \\ \end{align*}$ In this example two types of pairs are used. Self-loop pairs (blue ones) Other pairs (red ones) Here is the matrix representation of such relation. Self-loops pairs selected out of $7$ gray places. Other pairs selected out of $\begin{align*} \frac{7^2-7}{2} = 21 \end{align*}$ green places. Other pairs will only be selected in even numbers. Like $\{1,2\},\{2,1\}$ Here two pairs. To select two Other pairs we can choose only one green place out of $21$. To select four Other pairs we can choose only two green places out of $21$ and so on.. dd answered Feb 19, 2017 selected Feb 20, 2017 by Arnabi dd comment Share Follow See all 3 Comments See all 3 3 Comments reply dd commented Feb 19, 2017 reply Follow Share may be wrong please verify ! 0 votes 0 votes Arnabi commented Feb 20, 2017 reply Follow Share Thanks debashish ...nicely explained..got it ..:) 1 votes 1 votes Akriti sood commented Feb 28, 2017 reply Follow Share why cant i solve it like this - selecting 4 numbers from 7 and then finding symmetric relations from these 4 numbers. i.e 7C4 * [2n * (n2 - n)/2] because symmetric relation has ordered paris as (1,3) and (3,1) are different. 1 votes 1 votes Please log in or register to add a comment.