Generalized Distributive Law
$begingroup$
This is problem 3.11, p.29, Introduction to Set Theory, Hrbacek and Jech.
$$
big( bigcap_{a in A} F_a big) cup
big( bigcap_{b in B} G_b big)
= bigcap_{(a,b) in A times B} (F_a cup G_b).
$$
It is my attempt.
$
begin{equation}
x in (LHS) \
Leftrightarrow x in bigcap_{a in A} F_a ~~ or ~~ x in bigcap_{b in B} G_b \
Leftrightarrow (x in F_a ~~for ~~all~~ain A)~~or~~
(xin G_b~~ for~~all~~bin B).
end{equation}
$
And,
$
begin{equation}
xin(RHS) \
Leftrightarrow x in F_a cup G_b~~for ~~all~~(a,b) in A times B \
Leftrightarrow (xin F_a~~or ~~ x in G_b)
~~for ~~all~~(a,b) in A times B.
end{equation}
$
I want to change the position of phrases in each last sentences, but I'm not sure doing it preserves if and only if condition.
Please help my problem.
set-theory
asked Dec 31 '18 at 11:36
Doyun NamDoyun Nam
67119
$endgroup$
add a comment |
$begingroup$
This is problem 3.11, p.29, Introduction to Set Theory, Hrbacek and Jech.
$$
big( bigcap_{a in A} F_a big) cup
big( bigcap_{b in B} G_b big)
= bigcap_{(a,b) in A times B} (F_a cup G_b).
$$
It is my attempt.
$
begin{equation}
x in (LHS) \
Leftrightarrow x in bigcap_{a in A} F_a ~~ or ~~ x in bigcap_{b in B} G_b \
Leftrightarrow (x in F_a ~~for ~~all~~ain A)~~or~~
(xin G_b~~ for~~all~~bin B).
end{equation}
$
And,
$
begin{equation}
xin(RHS) \
Leftrightarrow x in F_a cup G_b~~for ~~all~~(a,b) in A times B \
Leftrightarrow (xin F_a~~or ~~ x in G_b)
~~for ~~all~~(a,b) in A times B.
end{equation}
$
I want to change the position of phrases in each last sentences, but I'm not sure doing it preserves if and only if condition.
Please help my problem.
set-theory
asked Dec 31 '18 at 11:36
Doyun NamDoyun Nam
67119
$endgroup$
add a comment |
$begingroup$
This is problem 3.11, p.29, Introduction to Set Theory, Hrbacek and Jech.
$$
big( bigcap_{a in A} F_a big) cup
big( bigcap_{b in B} G_b big)
= bigcap_{(a,b) in A times B} (F_a cup G_b).
$$
It is my attempt.
$
begin{equation}
x in (LHS) \
Leftrightarrow x in bigcap_{a in A} F_a ~~ or ~~ x in bigcap_{b in B} G_b \
Leftrightarrow (x in F_a ~~for ~~all~~ain A)~~or~~
(xin G_b~~ for~~all~~bin B).
end{equation}
$
And,
$
begin{equation}
xin(RHS) \
Leftrightarrow x in F_a cup G_b~~for ~~all~~(a,b) in A times B \
Leftrightarrow (xin F_a~~or ~~ x in G_b)
~~for ~~all~~(a,b) in A times B.
end{equation}
$
I want to change the position of phrases in each last sentences, but I'm not sure doing it preserves if and only if condition.
Please help my problem.
set-theory
asked Dec 31 '18 at 11:36
Doyun NamDoyun Nam
67119
$endgroup$
This is problem 3.11, p.29, Introduction to Set Theory, Hrbacek and Jech.
$$
big( bigcap_{a in A} F_a big) cup
big( bigcap_{b in B} G_b big)
= bigcap_{(a,b) in A times B} (F_a cup G_b).
$$
It is my attempt.
$
begin{equation}
x in (LHS) \
Leftrightarrow x in bigcap_{a in A} F_a ~~ or ~~ x in bigcap_{b in B} G_b \
Leftrightarrow (x in F_a ~~for ~~all~~ain A)~~or~~
(xin G_b~~ for~~all~~bin B).
end{equation}
$
And,
$
begin{equation}
xin(RHS) \
Leftrightarrow x in F_a cup G_b~~for ~~all~~(a,b) in A times B \
Leftrightarrow (xin F_a~~or ~~ x in G_b)
~~for ~~all~~(a,b) in A times B.
end{equation}
$
I want to change the position of phrases in each last sentences, but I'm not sure doing it preserves if and only if condition.
Please help my problem.
set-theory
set-theory
asked Dec 31 '18 at 11:36
Doyun NamDoyun Nam
67119
asked Dec 31 '18 at 11:36
Doyun NamDoyun Nam
67119
asked Dec 31 '18 at 11:36
Doyun NamDoyun Nam
67119
asked Dec 31 '18 at 11:36
Doyun NamDoyun Nam
67119
asked Dec 31 '18 at 11:36
Doyun NamDoyun Nam
67119
67119
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
It is more handsome to do the second part by proving that: $$xnotintext{ LHS }implies xnotintext{ RHS }$$
Doing it your way the following statements are equivalent:
- $forall ain A;forall bin B;[xin F_avee xin G_b]$
- $forall ain A;[xin F_aveeforall bin B;[xin G_b]]$
- $forall bin B;[xin G_b]veeforall ain A;[xin F_a]$
This on base of the rule: $$forall x;[ Qvee P(x)]iff Qveeforall x;P(x)$$
answered Dec 31 '18 at 11:55
drhabdrhab
102k545136
$endgroup$
$begingroup$
I understand your explanation, thank you! When I studied this book, I met some logical confusion. Can you recommend a basic mathematical logic textbook to me for studying undergraduate-level set theory?
$endgroup$
– Doyun Nam
Dec 31 '18 at 12:06
1
$begingroup$
Glad to help. Sorry, but I am not familiar with any mathematical textbook on logic:-(.
$endgroup$
– drhab
Dec 31 '18 at 12:09
add a comment |
Your Answer
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
It is more handsome to do the second part by proving that: $$xnotintext{ LHS }implies xnotintext{ RHS }$$
Doing it your way the following statements are equivalent:
- $forall ain A;forall bin B;[xin F_avee xin G_b]$
- $forall ain A;[xin F_aveeforall bin B;[xin G_b]]$
- $forall bin B;[xin G_b]veeforall ain A;[xin F_a]$
This on base of the rule: $$forall x;[ Qvee P(x)]iff Qveeforall x;P(x)$$
answered Dec 31 '18 at 11:55
drhabdrhab
102k545136
$endgroup$
$begingroup$
I understand your explanation, thank you! When I studied this book, I met some logical confusion. Can you recommend a basic mathematical logic textbook to me for studying undergraduate-level set theory?
$endgroup$
– Doyun Nam
Dec 31 '18 at 12:06
1
$begingroup$
Glad to help. Sorry, but I am not familiar with any mathematical textbook on logic:-(.
$endgroup$
– drhab
Dec 31 '18 at 12:09
add a comment |
$begingroup$
It is more handsome to do the second part by proving that: $$xnotintext{ LHS }implies xnotintext{ RHS }$$
Doing it your way the following statements are equivalent:
- $forall ain A;forall bin B;[xin F_avee xin G_b]$
- $forall ain A;[xin F_aveeforall bin B;[xin G_b]]$
- $forall bin B;[xin G_b]veeforall ain A;[xin F_a]$
This on base of the rule: $$forall x;[ Qvee P(x)]iff Qveeforall x;P(x)$$
answered Dec 31 '18 at 11:55
drhabdrhab
102k545136
$endgroup$
$begingroup$
I understand your explanation, thank you! When I studied this book, I met some logical confusion. Can you recommend a basic mathematical logic textbook to me for studying undergraduate-level set theory?
$endgroup$
– Doyun Nam
Dec 31 '18 at 12:06
1
$begingroup$
Glad to help. Sorry, but I am not familiar with any mathematical textbook on logic:-(.
$endgroup$
– drhab
Dec 31 '18 at 12:09
add a comment |
$begingroup$
It is more handsome to do the second part by proving that: $$xnotintext{ LHS }implies xnotintext{ RHS }$$
Doing it your way the following statements are equivalent:
- $forall ain A;forall bin B;[xin F_avee xin G_b]$
- $forall ain A;[xin F_aveeforall bin B;[xin G_b]]$
- $forall bin B;[xin G_b]veeforall ain A;[xin F_a]$
This on base of the rule: $$forall x;[ Qvee P(x)]iff Qveeforall x;P(x)$$
answered Dec 31 '18 at 11:55
drhabdrhab
102k545136
$endgroup$
It is more handsome to do the second part by proving that: $$xnotintext{ LHS }implies xnotintext{ RHS }$$
Doing it your way the following statements are equivalent:
- $forall ain A;forall bin B;[xin F_avee xin G_b]$
- $forall ain A;[xin F_aveeforall bin B;[xin G_b]]$
- $forall bin B;[xin G_b]veeforall ain A;[xin F_a]$
This on base of the rule: $$forall x;[ Qvee P(x)]iff Qveeforall x;P(x)$$
answered Dec 31 '18 at 11:55
drhabdrhab
102k545136
answered Dec 31 '18 at 11:55
drhabdrhab
102k545136
answered Dec 31 '18 at 11:55
drhabdrhab
102k545136
answered Dec 31 '18 at 11:55
drhabdrhab
102k545136
102k545136
$begingroup$
I understand your explanation, thank you! When I studied this book, I met some logical confusion. Can you recommend a basic mathematical logic textbook to me for studying undergraduate-level set theory?
$endgroup$
– Doyun Nam
Dec 31 '18 at 12:06
1
$begingroup$
Glad to help. Sorry, but I am not familiar with any mathematical textbook on logic:-(.
$endgroup$
– drhab
Dec 31 '18 at 12:09
add a comment |
$begingroup$
I understand your explanation, thank you! When I studied this book, I met some logical confusion. Can you recommend a basic mathematical logic textbook to me for studying undergraduate-level set theory?
$endgroup$
– Doyun Nam
Dec 31 '18 at 12:06
1
$begingroup$
Glad to help. Sorry, but I am not familiar with any mathematical textbook on logic:-(.
$endgroup$
– drhab
Dec 31 '18 at 12:09
$begingroup$
I understand your explanation, thank you! When I studied this book, I met some logical confusion. Can you recommend a basic mathematical logic textbook to me for studying undergraduate-level set theory?
$endgroup$
– Doyun Nam
Dec 31 '18 at 12:06
$begingroup$
I understand your explanation, thank you! When I studied this book, I met some logical confusion. Can you recommend a basic mathematical logic textbook to me for studying undergraduate-level set theory?
$endgroup$
– Doyun Nam
Dec 31 '18 at 12:06
1
1
$begingroup$
Glad to help. Sorry, but I am not familiar with any mathematical textbook on logic:-(.
$endgroup$
– drhab
Dec 31 '18 at 12:09
$begingroup$
Glad to help. Sorry, but I am not familiar with any mathematical textbook on logic:-(.
$endgroup$
– drhab
Dec 31 '18 at 12:09
add a comment |
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