Saved Bookmarks
| 1. |
Let `X={1,2,3,4,5,6,7,8,9}.` Let R be a relation in `X` given by `R_1={(x,y):x-y` is divisible by `3}` and `R_2` another on `X` given by `R={(x,y):(x,y)uu{1,4,7}} or {x,y} uu {2,5,8} or {x,y} uu {3,6,9}}` Show that `R_1=R_2.` |
|
Answer» Here, `X = {1,2,3,4,5,6,7,8,9}` `R_1 = {(x,y): x-y` is divisible by `3}` As, `x-y` is divisible by `3`. `x-y = 3n` where `n in N` `=>x= y+3n` `y=1, x = 4` when `n=1` `y=1, x = 7` when `n=2` `y=4, x = 7` when `n=1` `y=4, x = 1` when `n=-1` `y=7, x = 4` when `n=-1` `y=7, x = 1` when `n=-2` `:. (x,y) = {(1,4),(1,7),(4,7),(4,1),(7,4),(7,1)}` `:.(x,y) sub {1,4,7}` Similarly, ` (x,y) = {(2,5),(2,8),(5,8),(5,2),(8,5),(8,2)}` `:.(x,y) sub {2,5,8}` Similarly, ` (x,y) = {(3,6),(3,9),(6,9),(6,3),(9,6),(9,3)}` `:.(x,y) sub {3,6,9}` `:. R_1 = {(x,y):(x,y) sub {1,4,7}} or {(x,y):(x,y) sub {2,5,8}} or {(x,y):(x,y) sub {3,6,9}}` We are given, `:. R_2 = {(x,y):(x,y) sub {1,4,7}} or {(x,y):(x,y) sub {2,5,8}} or {(x,y):(x,y) sub {3,6,9}}` `:. R_1 = R_2` |
|