sets theory-Realation proof question
$begingroup$
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
3,2841922
asked Jan 4 at 10:30
ga asga as
31
$endgroup$
add a comment |
$begingroup$
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
3,2841922
asked Jan 4 at 10:30
ga asga as
31
$endgroup$
1
$begingroup$
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.
$endgroup$
– Shaun
Jan 4 at 10:41
add a comment |
$begingroup$
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
3,2841922
asked Jan 4 at 10:30
ga asga as
31
$endgroup$
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
3,2841922
asked Jan 4 at 10:30
ga asga as
31
edited Jan 4 at 18:39
verret
3,2841922
asked Jan 4 at 10:30
ga asga as
31
edited Jan 4 at 18:39
verret
3,2841922
edited Jan 4 at 18:39
verret
3,2841922
edited Jan 4 at 18:39
verret
3,2841922
3,2841922
asked Jan 4 at 10:30
ga asga as
31
asked Jan 4 at 10:30
ga asga as
31
asked Jan 4 at 10:30
ga asga as
31
31
1
$begingroup$
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.
$endgroup$
– Shaun
Jan 4 at 10:41
add a comment |
1
$begingroup$
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.
$endgroup$
– Shaun
Jan 4 at 10:41
1
1
$begingroup$
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.
$endgroup$
– Shaun
Jan 4 at 10:41
$begingroup$
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.
$endgroup$
– Shaun
Jan 4 at 10:41
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
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
8,4022720
$endgroup$
$begingroup$
Many thanks! Helped me alot
$endgroup$
– ga as
Jan 4 at 12:43
$begingroup$
@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.
$endgroup$
– Cameron Buie
Jan 5 at 19:48
add a comment |
Your Answer
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1 Answer
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votes
$begingroup$
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
8,4022720
$endgroup$
$begingroup$
Many thanks! Helped me alot
$endgroup$
– ga as
Jan 4 at 12:43
$begingroup$
@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.
$endgroup$
– Cameron Buie
Jan 5 at 19:48
add a comment |
$begingroup$
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
8,4022720
$endgroup$
$begingroup$
Many thanks! Helped me alot
$endgroup$
– ga as
Jan 4 at 12:43
$begingroup$
@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.
$endgroup$
– Cameron Buie
Jan 5 at 19:48
add a comment |
$begingroup$
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
8,4022720
$endgroup$
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
8,4022720
answered Jan 4 at 12:07
William ElliotWilliam Elliot
8,4022720
answered Jan 4 at 12:07
William ElliotWilliam Elliot
8,4022720
answered Jan 4 at 12:07
William ElliotWilliam Elliot
8,4022720
8,4022720
$begingroup$
Many thanks! Helped me alot
$endgroup$
– ga as
Jan 4 at 12:43
$begingroup$
@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.
$endgroup$
– Cameron Buie
Jan 5 at 19:48
add a comment |
$begingroup$
Many thanks! Helped me alot
$endgroup$
– ga as
Jan 4 at 12:43
$begingroup$
@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.
$endgroup$
– Cameron Buie
Jan 5 at 19:48
$begingroup$
Many thanks! Helped me alot
$endgroup$
– ga as
Jan 4 at 12:43
$begingroup$
Many thanks! Helped me alot
$endgroup$
– ga as
Jan 4 at 12:43
$begingroup$
@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.
$endgroup$
– Cameron Buie
Jan 5 at 19:48
$begingroup$
@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.
$endgroup$
– Cameron Buie
Jan 5 at 19:48
add a comment |
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$begingroup$
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.
$endgroup$
– Shaun
Jan 4 at 10:41