sets theory-Realation proof question
Let $A,B,C$ and $R,S,T$ be sets.
And assume that
$$ R subseteq A times B, ~~ S subseteq B times C, ~~
T subseteq B times C.$$
Then, I want to show that
$$
begin{equation}
(S circ R) cap (T circ R) subseteq (Scap T) circ R, \
(S circ R) cup (T circ R) = (S cup T) circ R.
end{equation}
$$
I tried solving this for hours.
I have no idea to even handle this.
I would love any kind of help or assistence.
Thank you.
discrete-mathematics proof-explanation relations
edited Jan 4 at 18:39
verret
2,9941818
asked Jan 4 at 10:30
ga asga as
31
New contributor
ga as is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |
Let $A,B,C$ and $R,S,T$ be sets.
And assume that
$$ R subseteq A times B, ~~ S subseteq B times C, ~~
T subseteq B times C.$$
Then, I want to show that
$$
begin{equation}
(S circ R) cap (T circ R) subseteq (Scap T) circ R, \
(S circ R) cup (T circ R) = (S cup T) circ R.
end{equation}
$$
I tried solving this for hours.
I have no idea to even handle this.
I would love any kind of help or assistence.
Thank you.
discrete-mathematics proof-explanation relations
edited Jan 4 at 18:39
verret
2,9941818
asked Jan 4 at 10:30
ga asga as
31
New contributor
ga as is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be put on hold. To prevent that, please edit the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers.
– José Carlos Santos
Jan 4 at 10:35
1
What do you know about set theory already? What text is this problem from? What course are you taking? Providing information like this would be a suitable alternative to any ideas you have to solve the problem.
– Shaun
Jan 4 at 10:41
add a comment |
Let $A,B,C$ and $R,S,T$ be sets.
And assume that
$$ R subseteq A times B, ~~ S subseteq B times C, ~~
T subseteq B times C.$$
Then, I want to show that
$$
begin{equation}
(S circ R) cap (T circ R) subseteq (Scap T) circ R, \
(S circ R) cup (T circ R) = (S cup T) circ R.
end{equation}
$$
I tried solving this for hours.
I have no idea to even handle this.
I would love any kind of help or assistence.
Thank you.
discrete-mathematics proof-explanation relations
edited Jan 4 at 18:39
verret
2,9941818
asked Jan 4 at 10:30
ga asga as
31
New contributor
ga as is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Let $A,B,C$ and $R,S,T$ be sets.
And assume that
$$ R subseteq A times B, ~~ S subseteq B times C, ~~
T subseteq B times C.$$
Then, I want to show that
$$
begin{equation}
(S circ R) cap (T circ R) subseteq (Scap T) circ R, \
(S circ R) cup (T circ R) = (S cup T) circ R.
end{equation}
$$
I tried solving this for hours.
I have no idea to even handle this.
I would love any kind of help or assistence.
Thank you.
discrete-mathematics proof-explanation relations
discrete-mathematics proof-explanation relations
edited Jan 4 at 18:39
verret
2,9941818
asked Jan 4 at 10:30
ga asga as
31
New contributor
ga as is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited Jan 4 at 18:39
verret
2,9941818
asked Jan 4 at 10:30
ga asga as
31
New contributor
ga as is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited Jan 4 at 18:39
verret
2,9941818
edited Jan 4 at 18:39
verret
2,9941818
edited Jan 4 at 18:39
verret
2,9941818
2,9941818
asked Jan 4 at 10:30
ga asga as
31
New contributor
ga as is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Jan 4 at 10:30
ga asga as
31
asked Jan 4 at 10:30
ga asga as
31
31
New contributor
ga as is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
ga as is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
ga as is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be put on hold. To prevent that, please edit the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers.
– José Carlos Santos
Jan 4 at 10:35
1
What do you know about set theory already? What text is this problem from? What course are you taking? Providing information like this would be a suitable alternative to any ideas you have to solve the problem.
– Shaun
Jan 4 at 10:41
add a comment |
Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be put on hold. To prevent that, please edit the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers.
– José Carlos Santos
Jan 4 at 10:35
1
What do you know about set theory already? What text is this problem from? What course are you taking? Providing information like this would be a suitable alternative to any ideas you have to solve the problem.
– Shaun
Jan 4 at 10:41
Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be put on hold. To prevent that, please edit the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers.
– José Carlos Santos
Jan 4 at 10:35
Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be put on hold. To prevent that, please edit the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers.
– José Carlos Santos
Jan 4 at 10:35
1
1
What do you know about set theory already? What text is this problem from? What course are you taking? Providing information like this would be a suitable alternative to any ideas you have to solve the problem.
– Shaun
Jan 4 at 10:41
What do you know about set theory already? What text is this problem from? What course are you taking? Providing information like this would be a suitable alternative to any ideas you have to solve the problem.
– Shaun
Jan 4 at 10:41
add a comment |
1 Answer
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If a(R o (S $cap$ T))c, then exists
b in B with aRb and b(S $cap$ T)c.
Thus aRb, bSc and bTc; a(RoS)c, a(RoT)c.
In conclusion, R o (S $cap$ T) subset RoS $cap$ RoT.
a(R o (S $cup$ T))c iff exists b in B with
aRb, b(S $cup$ T)c iff aRb and (bSc or bTc)
iff (aRb and bSc) or (aRb and bTc)
iff a(RoS)c or a(RoT)c.
In conclusion R o (S $cup$ T) = RoS $cup$ RoT.
Note that I'm using the conventions aRb for (a,b) in R and
for composition RoS ={ (a,c) : exists b with (aRb and bSc) }
answered Jan 4 at 12:07
William ElliotWilliam Elliot
7,3912720
Many thanks! Helped me alot
– ga as
Jan 4 at 12:43
@William: You've been on the site long enough to have a general idea how to use MathJax, and not rely on the kindness of others to make your posts look good. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to. Please look into it.
– Cameron Buie
2 days ago
add a comment |
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1 Answer
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If a(R o (S $cap$ T))c, then exists
b in B with aRb and b(S $cap$ T)c.
Thus aRb, bSc and bTc; a(RoS)c, a(RoT)c.
In conclusion, R o (S $cap$ T) subset RoS $cap$ RoT.
a(R o (S $cup$ T))c iff exists b in B with
aRb, b(S $cup$ T)c iff aRb and (bSc or bTc)
iff (aRb and bSc) or (aRb and bTc)
iff a(RoS)c or a(RoT)c.
In conclusion R o (S $cup$ T) = RoS $cup$ RoT.
Note that I'm using the conventions aRb for (a,b) in R and
for composition RoS ={ (a,c) : exists b with (aRb and bSc) }
answered Jan 4 at 12:07
William ElliotWilliam Elliot
7,3912720
Many thanks! Helped me alot
– ga as
Jan 4 at 12:43
@William: You've been on the site long enough to have a general idea how to use MathJax, and not rely on the kindness of others to make your posts look good. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to. Please look into it.
– Cameron Buie
2 days ago
add a comment |
If a(R o (S $cap$ T))c, then exists
b in B with aRb and b(S $cap$ T)c.
Thus aRb, bSc and bTc; a(RoS)c, a(RoT)c.
In conclusion, R o (S $cap$ T) subset RoS $cap$ RoT.
a(R o (S $cup$ T))c iff exists b in B with
aRb, b(S $cup$ T)c iff aRb and (bSc or bTc)
iff (aRb and bSc) or (aRb and bTc)
iff a(RoS)c or a(RoT)c.
In conclusion R o (S $cup$ T) = RoS $cup$ RoT.
Note that I'm using the conventions aRb for (a,b) in R and
for composition RoS ={ (a,c) : exists b with (aRb and bSc) }
answered Jan 4 at 12:07
William ElliotWilliam Elliot
7,3912720
Many thanks! Helped me alot
– ga as
Jan 4 at 12:43
@William: You've been on the site long enough to have a general idea how to use MathJax, and not rely on the kindness of others to make your posts look good. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to. Please look into it.
– Cameron Buie
2 days ago
add a comment |
If a(R o (S $cap$ T))c, then exists
b in B with aRb and b(S $cap$ T)c.
Thus aRb, bSc and bTc; a(RoS)c, a(RoT)c.
In conclusion, R o (S $cap$ T) subset RoS $cap$ RoT.
a(R o (S $cup$ T))c iff exists b in B with
aRb, b(S $cup$ T)c iff aRb and (bSc or bTc)
iff (aRb and bSc) or (aRb and bTc)
iff a(RoS)c or a(RoT)c.
In conclusion R o (S $cup$ T) = RoS $cup$ RoT.
Note that I'm using the conventions aRb for (a,b) in R and
for composition RoS ={ (a,c) : exists b with (aRb and bSc) }
answered Jan 4 at 12:07
William ElliotWilliam Elliot
7,3912720
If a(R o (S $cap$ T))c, then exists
b in B with aRb and b(S $cap$ T)c.
Thus aRb, bSc and bTc; a(RoS)c, a(RoT)c.
In conclusion, R o (S $cap$ T) subset RoS $cap$ RoT.
a(R o (S $cup$ T))c iff exists b in B with
aRb, b(S $cup$ T)c iff aRb and (bSc or bTc)
iff (aRb and bSc) or (aRb and bTc)
iff a(RoS)c or a(RoT)c.
In conclusion R o (S $cup$ T) = RoS $cup$ RoT.
Note that I'm using the conventions aRb for (a,b) in R and
for composition RoS ={ (a,c) : exists b with (aRb and bSc) }
answered Jan 4 at 12:07
William ElliotWilliam Elliot
7,3912720
answered Jan 4 at 12:07
William ElliotWilliam Elliot
7,3912720
answered Jan 4 at 12:07
William ElliotWilliam Elliot
7,3912720
answered Jan 4 at 12:07
William ElliotWilliam Elliot
7,3912720
7,3912720
Many thanks! Helped me alot
– ga as
Jan 4 at 12:43
@William: You've been on the site long enough to have a general idea how to use MathJax, and not rely on the kindness of others to make your posts look good. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to. Please look into it.
– Cameron Buie
2 days ago
add a comment |
Many thanks! Helped me alot
– ga as
Jan 4 at 12:43
@William: You've been on the site long enough to have a general idea how to use MathJax, and not rely on the kindness of others to make your posts look good. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to. Please look into it.
– Cameron Buie
2 days ago
Many thanks! Helped me alot
– ga as
Jan 4 at 12:43
Many thanks! Helped me alot
– ga as
Jan 4 at 12:43
@William: You've been on the site long enough to have a general idea how to use MathJax, and not rely on the kindness of others to make your posts look good. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to. Please look into it.
– Cameron Buie
2 days ago
@William: You've been on the site long enough to have a general idea how to use MathJax, and not rely on the kindness of others to make your posts look good. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to. Please look into it.
– Cameron Buie
2 days ago
add a comment |
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ga as is a new contributor. Be nice, and check out our Code of Conduct.
ga as is a new contributor. Be nice, and check out our Code of Conduct.
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Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be put on hold. To prevent that, please edit the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers.
– José Carlos Santos
Jan 4 at 10:35
1
What do you know about set theory already? What text is this problem from? What course are you taking? Providing information like this would be a suitable alternative to any ideas you have to solve the problem.
– Shaun
Jan 4 at 10:41