Real Analysis

[Brainwave]

I want to prove this theorem but I don't know where to go about it. THEOREM: Let R be a relation from X to Y and S be a relation from Y to Z and T be a relation a relation from set Z to set Y. Then (I) (R^-1)^-1 = R (Ii) Dm(R^-1) = Rng(R) (Iii) Rng(R^-1) = Dm(R) (Iv) To(SoR) = (ToS)oR (V) (SoR)^-1 = R^-1 o S^-1. Pls I need the proof. Anyone you can help of.

 

[HallsofIvy]

I want to prove this theorem but I don't know where to go about it. THEOREM: Let R be a relation from X to Y and S be a relation from Y to Z and T be a relation a relation from set Z to set Y. Then (I) (R^-1)^-1 = R (Ii) Dm(R^-1) = Rng(R) (Iii) Rng(R^-1) = Dm(R) (Iv) To(SoR) = (ToS)oR (V) (SoR)^-1 = R^-1 o S^-1. Pls I need the proof. Anyone you can help of.

The proof of what? You have five different statements. Are you asking for a proof of each? (I) to begin with, puzzles me. That is (part of) what I would think of as the definition of "R^-1". If you are asked to prove it then what is your definition of R^-1?

All of these follow from the definition of R^-1 and if you are not using "R(R^-1)= R^-1(R)= I" as that definition then I do not know what definition you are using.

 

[Deleted member 4993]

I want to prove this theorem but I don't know where to go about it. THEOREM: Let R be a relation from X to Y and S be a relation from Y to Z and T be a relation a relation from set Z to set Y. Then (I) (R^-1)^-1 = R (Ii) Dm(R^-1) = Rng(R) (Iii) Rng(R^-1) = Dm(R) (Iv) To(SoR) = (ToS)oR (V) (SoR)^-1 = R^-1 o S^-1. Pls I need the proof. Anyone you can help of.

Please add line-breaks in your post to render readability!

 

[Brainwave]

To prove (I) to (v)

 

[stapel]

Please reformat for clarity and reply with the requested definitions, so that people have a chance at understanding what you are asking. When you reply, please show your work this time. Thank you!