Composition of Relations


by 1MileCrash
Tags: composition, relations
1MileCrash
1MileCrash is offline
#1
Apr30-12, 11:35 PM
1MileCrash's Avatar
P: 1,227
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.
Phys.Org News Partner Science news on Phys.org
Cougars' diverse diet helped them survive the Pleistocene mass extinction
Cyber risks can cause disruption on scale of 2008 crisis, study says
Mantis shrimp stronger than airplanes
HallsofIvy
HallsofIvy is offline
#2
May1-12, 07:25 AM
Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 38,896
Quote Quote by 1MileCrash View Post
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) }
So [itex]S^{-1}[/itex] "maps" 1 to 2 and [itex]R^{-1}[/itex] maps 2 to 1. Therefore [itex]R^{-1}oS^{-1}[/itex] maps 1 to 1 and contains the pair (1, 1).

[itex]R^{-1}[/itex] also maps 2 to 2 so [itex]R^{-1}oS^{-1}[/itex] 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
1MileCrash is offline
#3
May1-12, 08:59 AM
1MileCrash's Avatar
P: 1,227
So (3,3) is in the composition because we have (5,3) and (3,5)?

SammyS
SammyS is offline
#4
May1-12, 09:55 AM
Emeritus
Sci Advisor
HW Helper
PF Gold
P: 7,416

Composition of Relations


Quote Quote by 1MileCrash View Post
So (3,3) is in the composition because we have (5,3) and (3,5)?
(3, 3) is in [itex]\displaystyle R^{-1}\circ S^{-1}[/itex] because, (3, 5) is in [itex]S^{-1}[/itex] and (5, 3) is in [itex]R^{-1}\ .[/itex]
1MileCrash
1MileCrash is offline
#5
May1-12, 10:02 AM
1MileCrash's Avatar
P: 1,227
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
SammyS is offline
#6
May1-12, 12:16 PM
Emeritus
Sci Advisor
HW Helper
PF Gold
P: 7,416
Quote Quote by 1MileCrash View Post
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.


Register to reply

Related Discussions
Can you name this composition? Materials & Chemical Engineering 6
Gauss Composition? and a naive composition law Linear & Abstract Algebra 3
Relations bet. Groups, from Relations between Resp. Presentations. Linear & Abstract Algebra 1
phase flow is the one-parameter group of transformations Differential Geometry 6
Beginner's mathematical proof / composition of relations Calculus & Beyond Homework 11