Proving $Atimes(Bcup C)=(Atimes B)cup(Atimes C)$
1
$begingroup$
Prove that $Atimes(Bcup C)=(Atimes B)cup(Atimes C)$
My Try:
$(x,y)in Atimes(Bcup C) $
$xin A$ and $yin(Bcup C)$
$(xin A$ and $yin B)$ or $(xin A$ and $yin C)$
$(x,y)in Atimes B$ or $(x,y)in Atimes C$
$(x,y)in (Atimes B)cup(Atimes C)$
$(x,y)in (Atimes B)cup(Atimes C)$
So, I proved $Atimes(Bcap C)subset(Atimes B)cup(Atimes C)$
My question: Do I also need to prove $(Atimes B)cup(Atimes C)subset Atimes(Bcap C)?$
discrete-mathematics proof-verification elementary-set-theory logic
asked Dec 2 '18 at 19:08
user982787user982787
1117
$endgroup$
add a comment |
1
$begingroup$
Prove that $Atimes(Bcup C)=(Atimes B)cup(Atimes C)$
My Try:
$(x,y)in Atimes(Bcup C) $
$xin A$ and $yin(Bcup C)$
$(xin A$ and $yin B)$ or $(xin A$ and $yin C)$
$(x,y)in Atimes B$ or $(x,y)in Atimes C$
$(x,y)in (Atimes B)cup(Atimes C)$
$(x,y)in (Atimes B)cup(Atimes C)$
So, I proved $Atimes(Bcap C)subset(Atimes B)cup(Atimes C)$
My question: Do I also need to prove $(Atimes B)cup(Atimes C)subset Atimes(Bcap C)?$
discrete-mathematics proof-verification elementary-set-theory logic
asked Dec 2 '18 at 19:08
user982787user982787
1117
$endgroup$
1
$begingroup$
Of course you do
$endgroup$
– tommy1996q
Dec 2 '18 at 19:10
1
$begingroup$
Yes you do. To prove equality you have to prove $xin A iff xin B$, which means $xin A Rightarrow xin B wedge xin B Rightarrow xin A$
$endgroup$
– NL1992
Dec 2 '18 at 19:10
add a comment |
1
1
1
$begingroup$
Prove that $Atimes(Bcup C)=(Atimes B)cup(Atimes C)$
My Try:
$(x,y)in Atimes(Bcup C) $
$xin A$ and $yin(Bcup C)$
$(xin A$ and $yin B)$ or $(xin A$ and $yin C)$
$(x,y)in Atimes B$ or $(x,y)in Atimes C$
$(x,y)in (Atimes B)cup(Atimes C)$
$(x,y)in (Atimes B)cup(Atimes C)$
So, I proved $Atimes(Bcap C)subset(Atimes B)cup(Atimes C)$
My question: Do I also need to prove $(Atimes B)cup(Atimes C)subset Atimes(Bcap C)?$
discrete-mathematics proof-verification elementary-set-theory logic
asked Dec 2 '18 at 19:08
user982787user982787
1117
$endgroup$
Prove that $Atimes(Bcup C)=(Atimes B)cup(Atimes C)$
My Try:
$(x,y)in Atimes(Bcup C) $
$xin A$ and $yin(Bcup C)$
$(xin A$ and $yin B)$ or $(xin A$ and $yin C)$
$(x,y)in Atimes B$ or $(x,y)in Atimes C$
$(x,y)in (Atimes B)cup(Atimes C)$
$(x,y)in (Atimes B)cup(Atimes C)$
So, I proved $Atimes(Bcap C)subset(Atimes B)cup(Atimes C)$
My question: Do I also need to prove $(Atimes B)cup(Atimes C)subset Atimes(Bcap C)?$
discrete-mathematics proof-verification elementary-set-theory logic
discrete-mathematics proof-verification elementary-set-theory logic
asked Dec 2 '18 at 19:08
user982787user982787
1117
asked Dec 2 '18 at 19:08
user982787user982787
1117
asked Dec 2 '18 at 19:08
user982787user982787
1117
asked Dec 2 '18 at 19:08
user982787user982787
1117
asked Dec 2 '18 at 19:08
user982787user982787
1117
1117
1
$begingroup$
Of course you do
$endgroup$
– tommy1996q
Dec 2 '18 at 19:10
1
$begingroup$
Yes you do. To prove equality you have to prove $xin A iff xin B$, which means $xin A Rightarrow xin B wedge xin B Rightarrow xin A$
$endgroup$
– NL1992
Dec 2 '18 at 19:10
add a comment |
1
$begingroup$
Of course you do
$endgroup$
– tommy1996q
Dec 2 '18 at 19:10
1
$begingroup$
Yes you do. To prove equality you have to prove $xin A iff xin B$, which means $xin A Rightarrow xin B wedge xin B Rightarrow xin A$
$endgroup$
– NL1992
Dec 2 '18 at 19:10
1
1
$begingroup$
Of course you do
$endgroup$
– tommy1996q
Dec 2 '18 at 19:10
$begingroup$
Of course you do
$endgroup$
– tommy1996q
Dec 2 '18 at 19:10
1
1
$begingroup$
Yes you do. To prove equality you have to prove $xin A iff xin B$, which means $xin A Rightarrow xin B wedge xin B Rightarrow xin A$
$endgroup$
– NL1992
Dec 2 '18 at 19:10
$begingroup$
Yes you do. To prove equality you have to prove $xin A iff xin B$, which means $xin A Rightarrow xin B wedge xin B Rightarrow xin A$
$endgroup$
– NL1992
Dec 2 '18 at 19:10
add a comment |
1 Answer
1
active
oldest
votes
3
$begingroup$
Just write $iff$ arrows between the statements.
answered Dec 2 '18 at 19:10
J.G.J.G.
24.4k22539
$endgroup$
$begingroup$
Where and what is it meant to write $iff$ ?
$endgroup$
– user982787
Dec 2 '18 at 19:30
1
$begingroup$
@user982787 You need to emphasize each of your lines is true iff the other is, so you can just place $iff$ at the beginning of each line (except the first one).
$endgroup$
– J.G.
Dec 2 '18 at 19:32
$begingroup$
If I place $iff$, then I need not prove $(A×B)∪(A×C)⊂A×(B∩C)$ this right?
$endgroup$
– user982787
Dec 2 '18 at 19:36
2
$begingroup$
@user982787 When as in a proof like this all inferences can be made validly in both directions, you get both proof directions for the price of one write-up.
$endgroup$
– J.G.
Dec 2 '18 at 19:59
$begingroup$
Just add "... and vice versa, as all implications are biconditional."
$endgroup$
– Graham Kemp
Dec 2 '18 at 21:40
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
3
$begingroup$
Just write $iff$ arrows between the statements.
answered Dec 2 '18 at 19:10
J.G.J.G.
24.4k22539
$endgroup$
$begingroup$
Where and what is it meant to write $iff$ ?
$endgroup$
– user982787
Dec 2 '18 at 19:30
1
$begingroup$
@user982787 You need to emphasize each of your lines is true iff the other is, so you can just place $iff$ at the beginning of each line (except the first one).
$endgroup$
– J.G.
Dec 2 '18 at 19:32
$begingroup$
If I place $iff$, then I need not prove $(A×B)∪(A×C)⊂A×(B∩C)$ this right?
$endgroup$
– user982787
Dec 2 '18 at 19:36
2
$begingroup$
@user982787 When as in a proof like this all inferences can be made validly in both directions, you get both proof directions for the price of one write-up.
$endgroup$
– J.G.
Dec 2 '18 at 19:59
$begingroup$
Just add "... and vice versa, as all implications are biconditional."
$endgroup$
– Graham Kemp
Dec 2 '18 at 21:40
add a comment |
3
$begingroup$
Just write $iff$ arrows between the statements.
answered Dec 2 '18 at 19:10
J.G.J.G.
24.4k22539
$endgroup$
$begingroup$
Where and what is it meant to write $iff$ ?
$endgroup$
– user982787
Dec 2 '18 at 19:30
1
$begingroup$
@user982787 You need to emphasize each of your lines is true iff the other is, so you can just place $iff$ at the beginning of each line (except the first one).
$endgroup$
– J.G.
Dec 2 '18 at 19:32
$begingroup$
If I place $iff$, then I need not prove $(A×B)∪(A×C)⊂A×(B∩C)$ this right?
$endgroup$
– user982787
Dec 2 '18 at 19:36
2
$begingroup$
@user982787 When as in a proof like this all inferences can be made validly in both directions, you get both proof directions for the price of one write-up.
$endgroup$
– J.G.
Dec 2 '18 at 19:59
$begingroup$
Just add "... and vice versa, as all implications are biconditional."
$endgroup$
– Graham Kemp
Dec 2 '18 at 21:40
add a comment |
3
3
3
$begingroup$
Just write $iff$ arrows between the statements.
answered Dec 2 '18 at 19:10
J.G.J.G.
24.4k22539
$endgroup$
Just write $iff$ arrows between the statements.
answered Dec 2 '18 at 19:10
J.G.J.G.
24.4k22539
answered Dec 2 '18 at 19:10
J.G.J.G.
24.4k22539
answered Dec 2 '18 at 19:10
J.G.J.G.
24.4k22539
answered Dec 2 '18 at 19:10
J.G.J.G.
24.4k22539
24.4k22539
$begingroup$
Where and what is it meant to write $iff$ ?
$endgroup$
– user982787
Dec 2 '18 at 19:30
1
$begingroup$
@user982787 You need to emphasize each of your lines is true iff the other is, so you can just place $iff$ at the beginning of each line (except the first one).
$endgroup$
– J.G.
Dec 2 '18 at 19:32
$begingroup$
If I place $iff$, then I need not prove $(A×B)∪(A×C)⊂A×(B∩C)$ this right?
$endgroup$
– user982787
Dec 2 '18 at 19:36
2
$begingroup$
@user982787 When as in a proof like this all inferences can be made validly in both directions, you get both proof directions for the price of one write-up.
$endgroup$
– J.G.
Dec 2 '18 at 19:59
$begingroup$
Just add "... and vice versa, as all implications are biconditional."
$endgroup$
– Graham Kemp
Dec 2 '18 at 21:40
add a comment |
$begingroup$
Where and what is it meant to write $iff$ ?
$endgroup$
– user982787
Dec 2 '18 at 19:30
1
$begingroup$
@user982787 You need to emphasize each of your lines is true iff the other is, so you can just place $iff$ at the beginning of each line (except the first one).
$endgroup$
– J.G.
Dec 2 '18 at 19:32
$begingroup$
If I place $iff$, then I need not prove $(A×B)∪(A×C)⊂A×(B∩C)$ this right?
$endgroup$
– user982787
Dec 2 '18 at 19:36
2
$begingroup$
@user982787 When as in a proof like this all inferences can be made validly in both directions, you get both proof directions for the price of one write-up.
$endgroup$
– J.G.
Dec 2 '18 at 19:59
$begingroup$
Just add "... and vice versa, as all implications are biconditional."
$endgroup$
– Graham Kemp
Dec 2 '18 at 21:40
$begingroup$
Where and what is it meant to write $iff$ ?
$endgroup$
– user982787
Dec 2 '18 at 19:30
$begingroup$
Where and what is it meant to write $iff$ ?
$endgroup$
– user982787
Dec 2 '18 at 19:30
1
1
$begingroup$
@user982787 You need to emphasize each of your lines is true iff the other is, so you can just place $iff$ at the beginning of each line (except the first one).
$endgroup$
– J.G.
Dec 2 '18 at 19:32
$begingroup$
@user982787 You need to emphasize each of your lines is true iff the other is, so you can just place $iff$ at the beginning of each line (except the first one).
$endgroup$
– J.G.
Dec 2 '18 at 19:32
$begingroup$
If I place $iff$, then I need not prove $(A×B)∪(A×C)⊂A×(B∩C)$ this right?
$endgroup$
– user982787
Dec 2 '18 at 19:36
$begingroup$
If I place $iff$, then I need not prove $(A×B)∪(A×C)⊂A×(B∩C)$ this right?
$endgroup$
– user982787
Dec 2 '18 at 19:36
2
2
$begingroup$
@user982787 When as in a proof like this all inferences can be made validly in both directions, you get both proof directions for the price of one write-up.
$endgroup$
– J.G.
Dec 2 '18 at 19:59
$begingroup$
@user982787 When as in a proof like this all inferences can be made validly in both directions, you get both proof directions for the price of one write-up.
$endgroup$
– J.G.
Dec 2 '18 at 19:59
$begingroup$
Just add "... and vice versa, as all implications are biconditional."
$endgroup$
– Graham Kemp
Dec 2 '18 at 21:40
$begingroup$
Just add "... and vice versa, as all implications are biconditional."
$endgroup$
– Graham Kemp
Dec 2 '18 at 21:40
add a comment |
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1
$begingroup$
Of course you do
$endgroup$
– tommy1996q
Dec 2 '18 at 19:10
1
$begingroup$
Yes you do. To prove equality you have to prove $xin A iff xin B$, which means $xin A Rightarrow xin B wedge xin B Rightarrow xin A$
$endgroup$
– NL1992
Dec 2 '18 at 19:10