Direct proof of Universal Set [discrete mathematics] The Next CEO of Stack OverflowDirect...
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Direct proof of Universal Set [discrete mathematics]
The Next CEO of Stack OverflowDirect proof set theory.Direct proof of empty set being subset of every setSubset proper subset proof help in discrete mathLet n be a perfect square. Use direct proof to show that n-1 is composite.Discrete Math Modulus Beginner Direct ProofProof by Cases [discrete mathematics]Draw a pentagon with any 2 angle sums being less than $216^o$Indirect Proof [discrete mathematics]Big-$O$ verification [discrete mathematics]Universal set proof [discrete mathematics]
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So I've come across this interesting proof question in my discrete mathematics textbook that I'm trying to solve for practice but unfortunately it does not have a solution. Any help with a reasonable solution and explanation so that I can verify my work would be greatly appreciated:
Suppose that $U$ is the universal set, and that $A$ and $B$ are two
arbitrary sets of elements of $U$:
a. First, suppose that $A$ contains at least two elements. Using a
direct proof, show that if every proper subset of $A$ is a subset of
$B$, then $A$ is a subset of $B$.
b. Give an example that shows that the implication from part (a) is
False if $A$ contains only one element.
discrete-mathematics proof-verification elementary-set-theory proof-writing proof-explanation
edited Mar 17 at 12:48
Andrés E. Caicedo
65.8k8160251
asked Mar 16 at 18:35
Joe BidenJoe Biden
65
$endgroup$
add a comment |
$begingroup$
So I've come across this interesting proof question in my discrete mathematics textbook that I'm trying to solve for practice but unfortunately it does not have a solution. Any help with a reasonable solution and explanation so that I can verify my work would be greatly appreciated:
Suppose that $U$ is the universal set, and that $A$ and $B$ are two
arbitrary sets of elements of $U$:
a. First, suppose that $A$ contains at least two elements. Using a
direct proof, show that if every proper subset of $A$ is a subset of
$B$, then $A$ is a subset of $B$.
b. Give an example that shows that the implication from part (a) is
False if $A$ contains only one element.
discrete-mathematics proof-verification elementary-set-theory proof-writing proof-explanation
edited Mar 17 at 12:48
Andrés E. Caicedo
65.8k8160251
asked Mar 16 at 18:35
Joe BidenJoe Biden
65
$endgroup$
1
$begingroup$
For the first one, just assume $A={p,q};pne q$. Given ${p},{q}subseteq Bimplies p,qin Bimplies Asubseteq B$. For the second, note that the only proper subset of a singleton set is the null set, which is a subset of every set $B$. You can substantiate using $A={1},B={2}$.
$endgroup$
– Shubham Johri
Mar 16 at 18:40
$begingroup$
Also, welcome to the website. You are advised to include some of your work and attempts at solving the problem, otherwise your post might get downvoted or closed.
$endgroup$
– Shubham Johri
Mar 16 at 18:45
add a comment |
$begingroup$
So I've come across this interesting proof question in my discrete mathematics textbook that I'm trying to solve for practice but unfortunately it does not have a solution. Any help with a reasonable solution and explanation so that I can verify my work would be greatly appreciated:
Suppose that $U$ is the universal set, and that $A$ and $B$ are two
arbitrary sets of elements of $U$:
a. First, suppose that $A$ contains at least two elements. Using a
direct proof, show that if every proper subset of $A$ is a subset of
$B$, then $A$ is a subset of $B$.
b. Give an example that shows that the implication from part (a) is
False if $A$ contains only one element.
discrete-mathematics proof-verification elementary-set-theory proof-writing proof-explanation
edited Mar 17 at 12:48
Andrés E. Caicedo
65.8k8160251
asked Mar 16 at 18:35
Joe BidenJoe Biden
65
$endgroup$
So I've come across this interesting proof question in my discrete mathematics textbook that I'm trying to solve for practice but unfortunately it does not have a solution. Any help with a reasonable solution and explanation so that I can verify my work would be greatly appreciated:
Suppose that $U$ is the universal set, and that $A$ and $B$ are two
arbitrary sets of elements of $U$:
a. First, suppose that $A$ contains at least two elements. Using a
direct proof, show that if every proper subset of $A$ is a subset of
$B$, then $A$ is a subset of $B$.
b. Give an example that shows that the implication from part (a) is
False if $A$ contains only one element.
discrete-mathematics proof-verification elementary-set-theory proof-writing proof-explanation
discrete-mathematics proof-verification elementary-set-theory proof-writing proof-explanation
edited Mar 17 at 12:48
Andrés E. Caicedo
65.8k8160251
asked Mar 16 at 18:35
Joe BidenJoe Biden
65
edited Mar 17 at 12:48
Andrés E. Caicedo
65.8k8160251
asked Mar 16 at 18:35
Joe BidenJoe Biden
65
edited Mar 17 at 12:48
Andrés E. Caicedo
65.8k8160251
edited Mar 17 at 12:48
Andrés E. Caicedo
65.8k8160251
edited Mar 17 at 12:48
Andrés E. Caicedo
65.8k8160251
65.8k8160251
asked Mar 16 at 18:35
Joe BidenJoe Biden
65
asked Mar 16 at 18:35
Joe BidenJoe Biden
65
asked Mar 16 at 18:35
Joe BidenJoe Biden
65
65
1
$begingroup$
For the first one, just assume $A={p,q};pne q$. Given ${p},{q}subseteq Bimplies p,qin Bimplies Asubseteq B$. For the second, note that the only proper subset of a singleton set is the null set, which is a subset of every set $B$. You can substantiate using $A={1},B={2}$.
$endgroup$
– Shubham Johri
Mar 16 at 18:40
$begingroup$
Also, welcome to the website. You are advised to include some of your work and attempts at solving the problem, otherwise your post might get downvoted or closed.
$endgroup$
– Shubham Johri
Mar 16 at 18:45
add a comment |
1
$begingroup$
For the first one, just assume $A={p,q};pne q$. Given ${p},{q}subseteq Bimplies p,qin Bimplies Asubseteq B$. For the second, note that the only proper subset of a singleton set is the null set, which is a subset of every set $B$. You can substantiate using $A={1},B={2}$.
$endgroup$
– Shubham Johri
Mar 16 at 18:40
$begingroup$
Also, welcome to the website. You are advised to include some of your work and attempts at solving the problem, otherwise your post might get downvoted or closed.
$endgroup$
– Shubham Johri
Mar 16 at 18:45
1
1
$begingroup$
For the first one, just assume $A={p,q};pne q$. Given ${p},{q}subseteq Bimplies p,qin Bimplies Asubseteq B$. For the second, note that the only proper subset of a singleton set is the null set, which is a subset of every set $B$. You can substantiate using $A={1},B={2}$.
$endgroup$
– Shubham Johri
Mar 16 at 18:40
$begingroup$
For the first one, just assume $A={p,q};pne q$. Given ${p},{q}subseteq Bimplies p,qin Bimplies Asubseteq B$. For the second, note that the only proper subset of a singleton set is the null set, which is a subset of every set $B$. You can substantiate using $A={1},B={2}$.
$endgroup$
– Shubham Johri
Mar 16 at 18:40
$begingroup$
Also, welcome to the website. You are advised to include some of your work and attempts at solving the problem, otherwise your post might get downvoted or closed.
$endgroup$
– Shubham Johri
Mar 16 at 18:45
$begingroup$
Also, welcome to the website. You are advised to include some of your work and attempts at solving the problem, otherwise your post might get downvoted or closed.
$endgroup$
– Shubham Johri
Mar 16 at 18:45
add a comment |
0
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$begingroup$
For the first one, just assume $A={p,q};pne q$. Given ${p},{q}subseteq Bimplies p,qin Bimplies Asubseteq B$. For the second, note that the only proper subset of a singleton set is the null set, which is a subset of every set $B$. You can substantiate using $A={1},B={2}$.
$endgroup$
– Shubham Johri
Mar 16 at 18:40
$begingroup$
Also, welcome to the website. You are advised to include some of your work and attempts at solving the problem, otherwise your post might get downvoted or closed.
$endgroup$
– Shubham Johri
Mar 16 at 18:45