Proving $Atimes(Bcup C)=(Atimes B)cup(Atimes C)$
up vote
1
down vote
favorite
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 at 19:08
user982787
897
add a comment |
up vote
1
down vote
favorite
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 at 19:08
user982787
897
1
Of course you do
– tommy1996q
Dec 2 at 19:10
1
Yes you do. To prove equality you have to prove $xin A iff xin B$, which means $xin A Rightarrow xin B wedge xin B Rightarrow xin A$
– NL1992
Dec 2 at 19:10
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
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 at 19:08
user982787
897
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 at 19:08
user982787
897
asked Dec 2 at 19:08
user982787
897
asked Dec 2 at 19:08
user982787
897
asked Dec 2 at 19:08
user982787
897
asked Dec 2 at 19:08
user982787
897
897
1
Of course you do
– tommy1996q
Dec 2 at 19:10
1
Yes you do. To prove equality you have to prove $xin A iff xin B$, which means $xin A Rightarrow xin B wedge xin B Rightarrow xin A$
– NL1992
Dec 2 at 19:10
add a comment |
1
Of course you do
– tommy1996q
Dec 2 at 19:10
1
Yes you do. To prove equality you have to prove $xin A iff xin B$, which means $xin A Rightarrow xin B wedge xin B Rightarrow xin A$
– NL1992
Dec 2 at 19:10
1
1
Of course you do
– tommy1996q
Dec 2 at 19:10
Of course you do
– tommy1996q
Dec 2 at 19:10
1
1
Yes you do. To prove equality you have to prove $xin A iff xin B$, which means $xin A Rightarrow xin B wedge xin B Rightarrow xin A$
– NL1992
Dec 2 at 19:10
Yes you do. To prove equality you have to prove $xin A iff xin B$, which means $xin A Rightarrow xin B wedge xin B Rightarrow xin A$
– NL1992
Dec 2 at 19:10
add a comment |
1 Answer
1
active
oldest
votes
up vote
3
down vote
Just write $iff$ arrows between the statements.
answered Dec 2 at 19:10
J.G.
20.5k21933
Where and what is it meant to write $iff$ ?
– user982787
Dec 2 at 19:30
1
@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).
– J.G.
Dec 2 at 19:32
If I place $iff$, then I need not prove $(A×B)∪(A×C)⊂A×(B∩C)$ this right?
– user982787
Dec 2 at 19:36
2
@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.
– J.G.
Dec 2 at 19:59
Just add "... and vice versa, as all implications are biconditional."
– Graham Kemp
Dec 2 at 21:40
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
3
down vote
Just write $iff$ arrows between the statements.
answered Dec 2 at 19:10
J.G.
20.5k21933
Where and what is it meant to write $iff$ ?
– user982787
Dec 2 at 19:30
1
@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).
– J.G.
Dec 2 at 19:32
If I place $iff$, then I need not prove $(A×B)∪(A×C)⊂A×(B∩C)$ this right?
– user982787
Dec 2 at 19:36
2
@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.
– J.G.
Dec 2 at 19:59
Just add "... and vice versa, as all implications are biconditional."
– Graham Kemp
Dec 2 at 21:40
add a comment |
up vote
3
down vote
Just write $iff$ arrows between the statements.
answered Dec 2 at 19:10
J.G.
20.5k21933
Where and what is it meant to write $iff$ ?
– user982787
Dec 2 at 19:30
1
@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).
– J.G.
Dec 2 at 19:32
If I place $iff$, then I need not prove $(A×B)∪(A×C)⊂A×(B∩C)$ this right?
– user982787
Dec 2 at 19:36
2
@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.
– J.G.
Dec 2 at 19:59
Just add "... and vice versa, as all implications are biconditional."
– Graham Kemp
Dec 2 at 21:40
add a comment |
up vote
3
down vote
up vote
3
down vote
Just write $iff$ arrows between the statements.
answered Dec 2 at 19:10
J.G.
20.5k21933
Just write $iff$ arrows between the statements.
answered Dec 2 at 19:10
J.G.
20.5k21933
answered Dec 2 at 19:10
J.G.
20.5k21933
answered Dec 2 at 19:10
J.G.
20.5k21933
answered Dec 2 at 19:10
J.G.
20.5k21933
20.5k21933
Where and what is it meant to write $iff$ ?
– user982787
Dec 2 at 19:30
1
@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).
– J.G.
Dec 2 at 19:32
If I place $iff$, then I need not prove $(A×B)∪(A×C)⊂A×(B∩C)$ this right?
– user982787
Dec 2 at 19:36
2
@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.
– J.G.
Dec 2 at 19:59
Just add "... and vice versa, as all implications are biconditional."
– Graham Kemp
Dec 2 at 21:40
add a comment |
Where and what is it meant to write $iff$ ?
– user982787
Dec 2 at 19:30
1
@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).
– J.G.
Dec 2 at 19:32
If I place $iff$, then I need not prove $(A×B)∪(A×C)⊂A×(B∩C)$ this right?
– user982787
Dec 2 at 19:36
2
@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.
– J.G.
Dec 2 at 19:59
Just add "... and vice versa, as all implications are biconditional."
– Graham Kemp
Dec 2 at 21:40
Where and what is it meant to write $iff$ ?
– user982787
Dec 2 at 19:30
Where and what is it meant to write $iff$ ?
– user982787
Dec 2 at 19:30
1
1
@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).
– J.G.
Dec 2 at 19:32
@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).
– J.G.
Dec 2 at 19:32
If I place $iff$, then I need not prove $(A×B)∪(A×C)⊂A×(B∩C)$ this right?
– user982787
Dec 2 at 19:36
If I place $iff$, then I need not prove $(A×B)∪(A×C)⊂A×(B∩C)$ this right?
– user982787
Dec 2 at 19:36
2
2
@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.
– J.G.
Dec 2 at 19:59
@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.
– J.G.
Dec 2 at 19:59
Just add "... and vice versa, as all implications are biconditional."
– Graham Kemp
Dec 2 at 21:40
Just add "... and vice versa, as all implications are biconditional."
– Graham Kemp
Dec 2 at 21:40
add a comment |
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1
Of course you do
– tommy1996q
Dec 2 at 19:10
1
Yes you do. To prove equality you have to prove $xin A iff xin B$, which means $xin A Rightarrow xin B wedge xin B Rightarrow xin A$
– NL1992
Dec 2 at 19:10