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__Question 1:__## Homework Statement

Prove:

For every x belongs to X, T

_{R∩S}(x) = T

_{R}(x) ∩ T

_{S}(x)

## Homework Equations

## The Attempt at a Solution

T

_{R}(x) = {x belongs to X such that <x,y> belongs to R}

T

_{S}(x) = {x belongs to X such that <x,y> belongs to S}

T

_{R∩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 T

_{R}(x) and T

_{S}(x)

Question 2: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.