# Homework Help: Composition of Relations

1. Apr 30, 2012

### 1MileCrash

1. The problem statement, all variables and given/known data

R = { (1,2), (3,5), (2,2), (2,5) }
S = { (2,1), (5,3), (5,1), (5,5) }

Explicitly find the relation R^-1 o S^-1

2. Relevant equations

3. The attempt at a solution

This was on my test.

First I just wrote down the inverses:

R^-1 = { (2,1), (5,3), (2,2), (5,2) }
S^-1 = { (1,2), (3,5), (1,5), (5,5) }

I didn't know what to do because the definition we learned defines 3 other sets, and all of the exercises in my test book has those 3 other sets defined.

For example, there are usually sets A, B, and C along with the sets R and S. So I have no idea how I can apply the definition to do this.

2. May 1, 2012

### HallsofIvy

So $S^{-1}$ "maps" 1 to 2 and $R^{-1}$ maps 2 to 1. Therefore $R^{-1}oS^{-1}$ maps 1 to 1 and contains the pair (1, 1).

$R^{-1}$ also maps 2 to 2 so $R^{-1}oS^{-1}$ also maps 1 to 2 and contains the pair (1, 2).

What 3 sets?

fog contains the pair (a, b) if and only if there exist some c such that g contains (a, c) and f contains (c, b).

Last edited by a moderator: May 1, 2012
3. May 1, 2012

### 1MileCrash

So (3,3) is in the composition because we have (5,3) and (3,5)?

4. May 1, 2012

### SammyS

Staff Emeritus
(3, 3) is in $\displaystyle R^{-1}\circ S^{-1}$ because, (3, 5) is in $S^{-1}$ and (5, 3) is in $R^{-1}\ .$

5. May 1, 2012

### 1MileCrash

I think the other three sets in my definition are A, B, and C and are dupposed to be the domain of R, the Range of R/domain of S, and the range of S.

Sound reasonable?

6. May 1, 2012

### SammyS

Staff Emeritus
As Halls said earlier, "What 3 sets?"

The domain of R is {1,2,3}.

The domain of S is {2,5}.

etc.