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Composition of Relations

by 1MileCrash
Tags: composition, relations
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1MileCrash
#1
Apr30-12, 11:35 PM
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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.
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HallsofIvy
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May1-12, 07:25 AM
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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
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May1-12, 08:59 AM
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So (3,3) is in the composition because we have (5,3) and (3,5)?

SammyS
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May1-12, 09:55 AM
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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
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May1-12, 10:02 AM
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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
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May1-12, 12:16 PM
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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.


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