Finding the Composition of Relations

[1MileCrash]

Homework Statement

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

Homework Equations

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.

 

[HallsofIvy]

Homework Statement

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

Homework Equations

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) }

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).

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.

What 3 sets?

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.

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).

 

[1MileCrash]

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

 

[SammyS]

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

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

 

[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?

 

[SammyS]

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

Sound reasonable?

As Halls said earlier, "What 3 sets?"

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

The domain of S is {2,5}.

etc.