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Discrete mathematics / set theory [closed]
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Claim: $A cup A' = U$ (where $U$ is the universal set)
How to prove this using set theory?
Through Venn diagram or through solving?
discrete-mathematics elementary-set-theory
edited Jul 30 at 15:06
Malkin
915421
asked Jul 30 at 14:20
Kavita Iyer
2
closed as off-topic by amWhy, José Carlos Santos, Shaun, Jyrki Lahtonen, Tyrone Aug 4 at 12:49
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." РamWhy, Jos̩ Carlos Santos, Shaun, Jyrki Lahtonen, Tyrone
add a comment |Â
up vote
-6
down vote
favorite
Claim: $A cup A' = U$ (where $U$ is the universal set)
How to prove this using set theory?
Through Venn diagram or through solving?
discrete-mathematics elementary-set-theory
edited Jul 30 at 15:06
Malkin
915421
asked Jul 30 at 14:20
Kavita Iyer
2
closed as off-topic by amWhy, José Carlos Santos, Shaun, Jyrki Lahtonen, Tyrone Aug 4 at 12:49
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." РamWhy, Jos̩ Carlos Santos, Shaun, Jyrki Lahtonen, Tyrone
Maybe using definitions ? What is set union ($cup$) ? And what is the complement of a set $A$ (i.e. $A'$) ?
– Mauro ALLEGRANZA
Jul 30 at 14:22
Venn diagrams are not formal proofs
– Goldname
Jul 30 at 14:24
Big hint: either a thing exists in this universe or it doesn't.
– Sean Roberson
Jul 30 at 14:25
add a comment |Â
up vote
-6
down vote
favorite
up vote
-6
down vote
favorite
Claim: $A cup A' = U$ (where $U$ is the universal set)
How to prove this using set theory?
Through Venn diagram or through solving?
discrete-mathematics elementary-set-theory
edited Jul 30 at 15:06
Malkin
915421
asked Jul 30 at 14:20
Kavita Iyer
2
Claim: $A cup A' = U$ (where $U$ is the universal set)
How to prove this using set theory?
Through Venn diagram or through solving?
discrete-mathematics elementary-set-theory
edited Jul 30 at 15:06
Malkin
915421
asked Jul 30 at 14:20
Kavita Iyer
2
edited Jul 30 at 15:06
Malkin
915421
edited Jul 30 at 15:06
Malkin
915421
edited Jul 30 at 15:06
Malkin
915421
915421
asked Jul 30 at 14:20
Kavita Iyer
2
asked Jul 30 at 14:20
Kavita Iyer
2
asked Jul 30 at 14:20
Kavita Iyer
2
2
closed as off-topic by amWhy, José Carlos Santos, Shaun, Jyrki Lahtonen, Tyrone Aug 4 at 12:49
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." РamWhy, Jos̩ Carlos Santos, Shaun, Jyrki Lahtonen, Tyrone
closed as off-topic by amWhy, José Carlos Santos, Shaun, Jyrki Lahtonen, Tyrone Aug 4 at 12:49
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." РamWhy, Jos̩ Carlos Santos, Shaun, Jyrki Lahtonen, Tyrone
Maybe using definitions ? What is set union ($cup$) ? And what is the complement of a set $A$ (i.e. $A'$) ?
– Mauro ALLEGRANZA
Jul 30 at 14:22
Venn diagrams are not formal proofs
– Goldname
Jul 30 at 14:24
Big hint: either a thing exists in this universe or it doesn't.
– Sean Roberson
Jul 30 at 14:25
add a comment |Â
Maybe using definitions ? What is set union ($cup$) ? And what is the complement of a set $A$ (i.e. $A'$) ?
– Mauro ALLEGRANZA
Jul 30 at 14:22
Venn diagrams are not formal proofs
– Goldname
Jul 30 at 14:24
Big hint: either a thing exists in this universe or it doesn't.
– Sean Roberson
Jul 30 at 14:25
Maybe using definitions ? What is set union ($cup$) ? And what is the complement of a set $A$ (i.e. $A'$) ?
– Mauro ALLEGRANZA
Jul 30 at 14:22
Maybe using definitions ? What is set union ($cup$) ? And what is the complement of a set $A$ (i.e. $A'$) ?
– Mauro ALLEGRANZA
Jul 30 at 14:22
Venn diagrams are not formal proofs
– Goldname
Jul 30 at 14:24
Venn diagrams are not formal proofs
– Goldname
Jul 30 at 14:24
Big hint: either a thing exists in this universe or it doesn't.
– Sean Roberson
Jul 30 at 14:25
Big hint: either a thing exists in this universe or it doesn't.
– Sean Roberson
Jul 30 at 14:25
add a comment |Â
2 Answers
2
active
oldest
votes
up vote
1
down vote
Formally, this does it (but is probably overkill):
Take $a in U$.
If $ain A$ then $ A subset (A cup A') implies ain A cup A'$.
If $a notin A$ then, by definition of $A'$, $ a in A'$. Then $A' subset (A cup A') implies ain A cup A'$.
Hence $U subset A cup A'$.
Conversely: since $A subset U$ and $A' subset U$ we immediately have $A cup A'subset U$.
Therefore $U = A cup A'$.
Here's a basic tip when proving that a set $A$ equals another set $B$: if $A subset B$ and $B subset A$ then $A = B$.
First take $a in A$ and prove that $a in B$. This gives you that $A subset B$. Do similar but backwards to show that $B subset A$.
answered Jul 30 at 14:43
Malkin
915421
1
nice formal proof for understanding of set theory without Venn diagram.
– Ninja hatori
Jul 30 at 15:51
add a comment |Â
up vote
0
down vote
x in U iff x in A or x not in A iff
x in A or x in A' iff x in A $cup$ A'.
answered Jul 31 at 3:30
William Elliot
5,0702414
add a comment |Â
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
Formally, this does it (but is probably overkill):
Take $a in U$.
If $ain A$ then $ A subset (A cup A') implies ain A cup A'$.
If $a notin A$ then, by definition of $A'$, $ a in A'$. Then $A' subset (A cup A') implies ain A cup A'$.
Hence $U subset A cup A'$.
Conversely: since $A subset U$ and $A' subset U$ we immediately have $A cup A'subset U$.
Therefore $U = A cup A'$.
Here's a basic tip when proving that a set $A$ equals another set $B$: if $A subset B$ and $B subset A$ then $A = B$.
First take $a in A$ and prove that $a in B$. This gives you that $A subset B$. Do similar but backwards to show that $B subset A$.
answered Jul 30 at 14:43
Malkin
915421
1
nice formal proof for understanding of set theory without Venn diagram.
– Ninja hatori
Jul 30 at 15:51
add a comment |Â
up vote
1
down vote
Formally, this does it (but is probably overkill):
Take $a in U$.
If $ain A$ then $ A subset (A cup A') implies ain A cup A'$.
If $a notin A$ then, by definition of $A'$, $ a in A'$. Then $A' subset (A cup A') implies ain A cup A'$.
Hence $U subset A cup A'$.
Conversely: since $A subset U$ and $A' subset U$ we immediately have $A cup A'subset U$.
Therefore $U = A cup A'$.
Here's a basic tip when proving that a set $A$ equals another set $B$: if $A subset B$ and $B subset A$ then $A = B$.
First take $a in A$ and prove that $a in B$. This gives you that $A subset B$. Do similar but backwards to show that $B subset A$.
answered Jul 30 at 14:43
Malkin
915421
1
nice formal proof for understanding of set theory without Venn diagram.
– Ninja hatori
Jul 30 at 15:51
add a comment |Â
up vote
1
down vote
up vote
1
down vote
Formally, this does it (but is probably overkill):
Take $a in U$.
If $ain A$ then $ A subset (A cup A') implies ain A cup A'$.
If $a notin A$ then, by definition of $A'$, $ a in A'$. Then $A' subset (A cup A') implies ain A cup A'$.
Hence $U subset A cup A'$.
Conversely: since $A subset U$ and $A' subset U$ we immediately have $A cup A'subset U$.
Therefore $U = A cup A'$.
Here's a basic tip when proving that a set $A$ equals another set $B$: if $A subset B$ and $B subset A$ then $A = B$.
First take $a in A$ and prove that $a in B$. This gives you that $A subset B$. Do similar but backwards to show that $B subset A$.
answered Jul 30 at 14:43
Malkin
915421
Formally, this does it (but is probably overkill):
Take $a in U$.
If $ain A$ then $ A subset (A cup A') implies ain A cup A'$.
If $a notin A$ then, by definition of $A'$, $ a in A'$. Then $A' subset (A cup A') implies ain A cup A'$.
Hence $U subset A cup A'$.
Conversely: since $A subset U$ and $A' subset U$ we immediately have $A cup A'subset U$.
Therefore $U = A cup A'$.
Here's a basic tip when proving that a set $A$ equals another set $B$: if $A subset B$ and $B subset A$ then $A = B$.
First take $a in A$ and prove that $a in B$. This gives you that $A subset B$. Do similar but backwards to show that $B subset A$.
answered Jul 30 at 14:43
Malkin
915421
edited Jul 30 at 14:56
answered Jul 30 at 14:43
Malkin
915421
answered Jul 30 at 14:43
Malkin
915421
answered Jul 30 at 14:43
Malkin
915421
915421
1
nice formal proof for understanding of set theory without Venn diagram.
– Ninja hatori
Jul 30 at 15:51
add a comment |Â
1
nice formal proof for understanding of set theory without Venn diagram.
– Ninja hatori
Jul 30 at 15:51
1
1
nice formal proof for understanding of set theory without Venn diagram.
– Ninja hatori
Jul 30 at 15:51
nice formal proof for understanding of set theory without Venn diagram.
– Ninja hatori
Jul 30 at 15:51
add a comment |Â
up vote
0
down vote
x in U iff x in A or x not in A iff
x in A or x in A' iff x in A $cup$ A'.
answered Jul 31 at 3:30
William Elliot
5,0702414
add a comment |Â
up vote
0
down vote
x in U iff x in A or x not in A iff
x in A or x in A' iff x in A $cup$ A'.
answered Jul 31 at 3:30
William Elliot
5,0702414
add a comment |Â
up vote
0
down vote
up vote
0
down vote
x in U iff x in A or x not in A iff
x in A or x in A' iff x in A $cup$ A'.
answered Jul 31 at 3:30
William Elliot
5,0702414
x in U iff x in A or x not in A iff
x in A or x in A' iff x in A $cup$ A'.
answered Jul 31 at 3:30
William Elliot
5,0702414
answered Jul 31 at 3:30
William Elliot
5,0702414
answered Jul 31 at 3:30
William Elliot
5,0702414
answered Jul 31 at 3:30
William Elliot
5,0702414
5,0702414
add a comment |Â
add a comment |Â
Maybe using definitions ? What is set union ($cup$) ? And what is the complement of a set $A$ (i.e. $A'$) ?
– Mauro ALLEGRANZA
Jul 30 at 14:22
Venn diagrams are not formal proofs
– Goldname
Jul 30 at 14:24
Big hint: either a thing exists in this universe or it doesn't.
– Sean Roberson
Jul 30 at 14:25