What is $ {square}_{mathscr A triangledown mathscr B} $?












2












$begingroup$


I'm currently working trough a set of notes on uniform spaces. I came upon a problem that I'm having some difficulties solving, so I am seeking assistance in the form of clarifications, hints or explanations.



In the notes I'm working trough, the following are the
definitions of the terms used:



• X is an arbitrary set;



• $ mathscr A $ and $ mathscr B $ are subsets of the power set of $ X $;



• $ {square}_{mathscr A} := {bigcup}_{A in mathscr A} A×A $;



• $ mathscr A triangledown mathscr B $ := {$A cup B | A in mathscr A , B in mathscr B , A cap B ne Ø$} ;



• $ circ $ is just composition of binary relations.



I am required to prove the following equality:



$ {square}_{mathscr A triangledown mathscr B} = ({square}_{mathscr A} circ {square}_{mathscr A}) cup ({square}_{mathscr A} circ {square}_{mathscr B}) cup ({square}_{mathscr B} circ {square}_{mathscr A}) cup ({square}_{mathscr B} circ {square}_{mathscr B}) $



Thus far, I have proved the inclusion



$ {square}_{mathscr A triangledown mathscr B} subseteq ({square}_{mathscr A} circ {square}_{mathscr A}) cup ({square}_{mathscr A} circ {square}_{mathscr B}) cup ({square}_{mathscr B} circ {square}_{mathscr A}) cup ({square}_{mathscr B} circ {square}_{mathscr B}) $



However, I'm having some trouble with the other direction. For instance, what if $ forall A in mathscr A, B in mathscr B (A cap B = Ø) $ ?



It seems to me thay in this case the left hand side is empty, whereas the right hand side is not, I suspect I may be missing some detail.






elementary-set-theory relations







share|cite|improve this question











$endgroup$












  • $begingroup$
    It's a notational mess. How can an element b, of A subset P(X×X) be a binary operation? If a,b are binary operations, what is their composition?
    $endgroup$
    – William Elliot
    Dec 2 '18 at 4:34












  • $begingroup$
    Not binary operations, binary relations.
    $endgroup$
    – Henno Brandsma
    Dec 2 '18 at 9:23










  • $begingroup$
    @WilliamElliot composition of relations, and it's defined as usual: if $R,S subseteq X times X$, then $R circ S = {(x,z) in X times X: exists y in X: (x,y) in R, (y,z) in S}$.
    $endgroup$
    – Henno Brandsma
    Dec 2 '18 at 9:25






  • 1




    $begingroup$
    The definition $Box_{mathscr{A}}$ is weird: $A in mathscr{A}$ means $A subseteq X times X$ and then $A times A$ is in $X^4$?
    $endgroup$
    – Henno Brandsma
    Dec 2 '18 at 9:34












  • $begingroup$
    Thanks for pointing that out, I had made some mistakes and typos in some of the definitions, I typed it up on my mobile phone and simply didn't notice. It should be correct now.
    $endgroup$
    – ToricTorus
    Dec 2 '18 at 17:13
















2












$begingroup$


I'm currently working trough a set of notes on uniform spaces. I came upon a problem that I'm having some difficulties solving, so I am seeking assistance in the form of clarifications, hints or explanations.



In the notes I'm working trough, the following are the
definitions of the terms used:



• X is an arbitrary set;



• $ mathscr A $ and $ mathscr B $ are subsets of the power set of $ X $;



• $ {square}_{mathscr A} := {bigcup}_{A in mathscr A} A×A $;



• $ mathscr A triangledown mathscr B $ := {$A cup B | A in mathscr A , B in mathscr B , A cap B ne Ø$} ;



• $ circ $ is just composition of binary relations.



I am required to prove the following equality:



$ {square}_{mathscr A triangledown mathscr B} = ({square}_{mathscr A} circ {square}_{mathscr A}) cup ({square}_{mathscr A} circ {square}_{mathscr B}) cup ({square}_{mathscr B} circ {square}_{mathscr A}) cup ({square}_{mathscr B} circ {square}_{mathscr B}) $



Thus far, I have proved the inclusion



$ {square}_{mathscr A triangledown mathscr B} subseteq ({square}_{mathscr A} circ {square}_{mathscr A}) cup ({square}_{mathscr A} circ {square}_{mathscr B}) cup ({square}_{mathscr B} circ {square}_{mathscr A}) cup ({square}_{mathscr B} circ {square}_{mathscr B}) $



However, I'm having some trouble with the other direction. For instance, what if $ forall A in mathscr A, B in mathscr B (A cap B = Ø) $ ?



It seems to me thay in this case the left hand side is empty, whereas the right hand side is not, I suspect I may be missing some detail.






elementary-set-theory relations







share|cite|improve this question











$endgroup$












  • $begingroup$
    It's a notational mess. How can an element b, of A subset P(X×X) be a binary operation? If a,b are binary operations, what is their composition?
    $endgroup$
    – William Elliot
    Dec 2 '18 at 4:34












  • $begingroup$
    Not binary operations, binary relations.
    $endgroup$
    – Henno Brandsma
    Dec 2 '18 at 9:23










  • $begingroup$
    @WilliamElliot composition of relations, and it's defined as usual: if $R,S subseteq X times X$, then $R circ S = {(x,z) in X times X: exists y in X: (x,y) in R, (y,z) in S}$.
    $endgroup$
    – Henno Brandsma
    Dec 2 '18 at 9:25






  • 1




    $begingroup$
    The definition $Box_{mathscr{A}}$ is weird: $A in mathscr{A}$ means $A subseteq X times X$ and then $A times A$ is in $X^4$?
    $endgroup$
    – Henno Brandsma
    Dec 2 '18 at 9:34












  • $begingroup$
    Thanks for pointing that out, I had made some mistakes and typos in some of the definitions, I typed it up on my mobile phone and simply didn't notice. It should be correct now.
    $endgroup$
    – ToricTorus
    Dec 2 '18 at 17:13














2












2








2


0



$begingroup$


I'm currently working trough a set of notes on uniform spaces. I came upon a problem that I'm having some difficulties solving, so I am seeking assistance in the form of clarifications, hints or explanations.



In the notes I'm working trough, the following are the
definitions of the terms used:



• X is an arbitrary set;



• $ mathscr A $ and $ mathscr B $ are subsets of the power set of $ X $;



• $ {square}_{mathscr A} := {bigcup}_{A in mathscr A} A×A $;



• $ mathscr A triangledown mathscr B $ := {$A cup B | A in mathscr A , B in mathscr B , A cap B ne Ø$} ;



• $ circ $ is just composition of binary relations.



I am required to prove the following equality:



$ {square}_{mathscr A triangledown mathscr B} = ({square}_{mathscr A} circ {square}_{mathscr A}) cup ({square}_{mathscr A} circ {square}_{mathscr B}) cup ({square}_{mathscr B} circ {square}_{mathscr A}) cup ({square}_{mathscr B} circ {square}_{mathscr B}) $



Thus far, I have proved the inclusion



$ {square}_{mathscr A triangledown mathscr B} subseteq ({square}_{mathscr A} circ {square}_{mathscr A}) cup ({square}_{mathscr A} circ {square}_{mathscr B}) cup ({square}_{mathscr B} circ {square}_{mathscr A}) cup ({square}_{mathscr B} circ {square}_{mathscr B}) $



However, I'm having some trouble with the other direction. For instance, what if $ forall A in mathscr A, B in mathscr B (A cap B = Ø) $ ?



It seems to me thay in this case the left hand side is empty, whereas the right hand side is not, I suspect I may be missing some detail.






elementary-set-theory relations







share|cite|improve this question











$endgroup$




I'm currently working trough a set of notes on uniform spaces. I came upon a problem that I'm having some difficulties solving, so I am seeking assistance in the form of clarifications, hints or explanations.



In the notes I'm working trough, the following are the
definitions of the terms used:



• X is an arbitrary set;



• $ mathscr A $ and $ mathscr B $ are subsets of the power set of $ X $;



• $ {square}_{mathscr A} := {bigcup}_{A in mathscr A} A×A $;



• $ mathscr A triangledown mathscr B $ := {$A cup B | A in mathscr A , B in mathscr B , A cap B ne Ø$} ;



• $ circ $ is just composition of binary relations.



I am required to prove the following equality:



$ {square}_{mathscr A triangledown mathscr B} = ({square}_{mathscr A} circ {square}_{mathscr A}) cup ({square}_{mathscr A} circ {square}_{mathscr B}) cup ({square}_{mathscr B} circ {square}_{mathscr A}) cup ({square}_{mathscr B} circ {square}_{mathscr B}) $



Thus far, I have proved the inclusion



$ {square}_{mathscr A triangledown mathscr B} subseteq ({square}_{mathscr A} circ {square}_{mathscr A}) cup ({square}_{mathscr A} circ {square}_{mathscr B}) cup ({square}_{mathscr B} circ {square}_{mathscr A}) cup ({square}_{mathscr B} circ {square}_{mathscr B}) $



However, I'm having some trouble with the other direction. For instance, what if $ forall A in mathscr A, B in mathscr B (A cap B = Ø) $ ?



It seems to me thay in this case the left hand side is empty, whereas the right hand side is not, I suspect I may be missing some detail.






elementary-set-theory relations




elementary-set-theory relations






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 2 '18 at 17:05







ToricTorus

















asked Dec 1 '18 at 23:35









ToricTorusToricTorus

204




204












  • $begingroup$
    It's a notational mess. How can an element b, of A subset P(X×X) be a binary operation? If a,b are binary operations, what is their composition?
    $endgroup$
    – William Elliot
    Dec 2 '18 at 4:34












  • $begingroup$
    Not binary operations, binary relations.
    $endgroup$
    – Henno Brandsma
    Dec 2 '18 at 9:23










  • $begingroup$
    @WilliamElliot composition of relations, and it's defined as usual: if $R,S subseteq X times X$, then $R circ S = {(x,z) in X times X: exists y in X: (x,y) in R, (y,z) in S}$.
    $endgroup$
    – Henno Brandsma
    Dec 2 '18 at 9:25






  • 1




    $begingroup$
    The definition $Box_{mathscr{A}}$ is weird: $A in mathscr{A}$ means $A subseteq X times X$ and then $A times A$ is in $X^4$?
    $endgroup$
    – Henno Brandsma
    Dec 2 '18 at 9:34












  • $begingroup$
    Thanks for pointing that out, I had made some mistakes and typos in some of the definitions, I typed it up on my mobile phone and simply didn't notice. It should be correct now.
    $endgroup$
    – ToricTorus
    Dec 2 '18 at 17:13


















  • $begingroup$
    It's a notational mess. How can an element b, of A subset P(X×X) be a binary operation? If a,b are binary operations, what is their composition?
    $endgroup$
    – William Elliot
    Dec 2 '18 at 4:34












  • $begingroup$
    Not binary operations, binary relations.
    $endgroup$
    – Henno Brandsma
    Dec 2 '18 at 9:23










  • $begingroup$
    @WilliamElliot composition of relations, and it's defined as usual: if $R,S subseteq X times X$, then $R circ S = {(x,z) in X times X: exists y in X: (x,y) in R, (y,z) in S}$.
    $endgroup$
    – Henno Brandsma
    Dec 2 '18 at 9:25






  • 1




    $begingroup$
    The definition $Box_{mathscr{A}}$ is weird: $A in mathscr{A}$ means $A subseteq X times X$ and then $A times A$ is in $X^4$?
    $endgroup$
    – Henno Brandsma
    Dec 2 '18 at 9:34












  • $begingroup$
    Thanks for pointing that out, I had made some mistakes and typos in some of the definitions, I typed it up on my mobile phone and simply didn't notice. It should be correct now.
    $endgroup$
    – ToricTorus
    Dec 2 '18 at 17:13
















$begingroup$
It's a notational mess. How can an element b, of A subset P(X×X) be a binary operation? If a,b are binary operations, what is their composition?
$endgroup$
– William Elliot
Dec 2 '18 at 4:34






$begingroup$
It's a notational mess. How can an element b, of A subset P(X×X) be a binary operation? If a,b are binary operations, what is their composition?
$endgroup$
– William Elliot
Dec 2 '18 at 4:34














$begingroup$
Not binary operations, binary relations.
$endgroup$
– Henno Brandsma
Dec 2 '18 at 9:23




$begingroup$
Not binary operations, binary relations.
$endgroup$
– Henno Brandsma
Dec 2 '18 at 9:23












$begingroup$
@WilliamElliot composition of relations, and it's defined as usual: if $R,S subseteq X times X$, then $R circ S = {(x,z) in X times X: exists y in X: (x,y) in R, (y,z) in S}$.
$endgroup$
– Henno Brandsma
Dec 2 '18 at 9:25




$begingroup$
@WilliamElliot composition of relations, and it's defined as usual: if $R,S subseteq X times X$, then $R circ S = {(x,z) in X times X: exists y in X: (x,y) in R, (y,z) in S}$.
$endgroup$
– Henno Brandsma
Dec 2 '18 at 9:25




1




1




$begingroup$
The definition $Box_{mathscr{A}}$ is weird: $A in mathscr{A}$ means $A subseteq X times X$ and then $A times A$ is in $X^4$?
$endgroup$
– Henno Brandsma
Dec 2 '18 at 9:34






$begingroup$
The definition $Box_{mathscr{A}}$ is weird: $A in mathscr{A}$ means $A subseteq X times X$ and then $A times A$ is in $X^4$?
$endgroup$
– Henno Brandsma
Dec 2 '18 at 9:34














$begingroup$
Thanks for pointing that out, I had made some mistakes and typos in some of the definitions, I typed it up on my mobile phone and simply didn't notice. It should be correct now.
$endgroup$
– ToricTorus
Dec 2 '18 at 17:13




$begingroup$
Thanks for pointing that out, I had made some mistakes and typos in some of the definitions, I typed it up on my mobile phone and simply didn't notice. It should be correct now.
$endgroup$
– ToricTorus
Dec 2 '18 at 17:13










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