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Ceci020
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I'm not sure if I did these 2 questions correctly, so would someone please check my work for any missing ideas or errors?
Question 1:
Prove:
For every x belongs to X, TR∩S(x) = TR(x) ∩ TS(x)
TR(x) = {x belongs to X such that <x,y> belongs to R}
TS(x) = {x belongs to X such that <x,y> belongs to S}
TR∩S(x) = {x belongs to X such that <x,y> belongs to R∩S}
<x,y> belongs to R∩S if <x,y> belongs to R and also belongs to S, which satisfy the definition above for TR(x) and TS(x)
Question 2:
R and S are equivalence relations over X
Prove R ∩ S is also an equivalence relation over X
Since R and S are equivalence relations over X, then for x in X, R and S satisfy properties:
Reflexive:
<x,x> belongs to R
<x,x> belongs to S
Symmetry:
<x,y> and <y,x> belong to R
<x,y> and <y,x> belong to S
Transitivity:
<x,y> belongs to R, <y,z> belongs to R; then <x,z> belongs to R
<x,y> belongs to S, <y,z> belongs to R, then <x,z> belongs to S
If R∩S is equivalence relation, then it must satisfy:
1/ <x,x> belongs to R∩S, meaning <x,x> belongs to R and also belongs to S
2/ <x,y> and <y,x> belong to R∩S, meaning <x,y> belongs to R and also belongs to S.
3/ <x,y> belongs to R∩S and <y,z> belongs to R∩S, then <x,z> belongs to R∩S, meaning <x,z> belongs to R and also belongs to S
All of these are satisfied by hypotheses.
So R∩S is equivalence relation over X.
Question 1:
Homework Statement
Prove:
For every x belongs to X, TR∩S(x) = TR(x) ∩ TS(x)
Homework Equations
The Attempt at a Solution
TR(x) = {x belongs to X such that <x,y> belongs to R}
TS(x) = {x belongs to X such that <x,y> belongs to S}
TR∩S(x) = {x belongs to X such that <x,y> belongs to R∩S}
<x,y> belongs to R∩S if <x,y> belongs to R and also belongs to S, which satisfy the definition above for TR(x) and TS(x)
Question 2:
Homework Statement
R and S are equivalence relations over X
Prove R ∩ S is also an equivalence relation over X
Homework Equations
The Attempt at a Solution
Since R and S are equivalence relations over X, then for x in X, R and S satisfy properties:
Reflexive:
<x,x> belongs to R
<x,x> belongs to S
Symmetry:
<x,y> and <y,x> belong to R
<x,y> and <y,x> belong to S
Transitivity:
<x,y> belongs to R, <y,z> belongs to R; then <x,z> belongs to R
<x,y> belongs to S, <y,z> belongs to R, then <x,z> belongs to S
If R∩S is equivalence relation, then it must satisfy:
1/ <x,x> belongs to R∩S, meaning <x,x> belongs to R and also belongs to S
2/ <x,y> and <y,x> belong to R∩S, meaning <x,y> belongs to R and also belongs to S.
3/ <x,y> belongs to R∩S and <y,z> belongs to R∩S, then <x,z> belongs to R∩S, meaning <x,z> belongs to R and also belongs to S
All of these are satisfied by hypotheses.
So R∩S is equivalence relation over X.