Proving function is bijective [closed]
up vote
-1
down vote
favorite
Show that if there is a function $g: B → A$, satisfying $g(f(x)) = x$ for all $x ∈ A$, and $f(g(x)) = x$ for all $x ∈ B$, then $f$ is a bijection, and inverse of $f = g$.
I'm not sure how to prove this.
algebra-precalculus
edited Nov 21 at 11:14
Robert Z
90.7k1057128
asked Nov 21 at 10:56
Patrick
335
closed as off-topic by José Carlos Santos, user302797, user26857, Christopher, Gibbs Nov 22 at 11:07
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." – José Carlos Santos, user302797, user26857, Christopher, Gibbs
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
up vote
-1
down vote
favorite
Show that if there is a function $g: B → A$, satisfying $g(f(x)) = x$ for all $x ∈ A$, and $f(g(x)) = x$ for all $x ∈ B$, then $f$ is a bijection, and inverse of $f = g$.
I'm not sure how to prove this.
algebra-precalculus
edited Nov 21 at 11:14
Robert Z
90.7k1057128
asked Nov 21 at 10:56
Patrick
335
closed as off-topic by José Carlos Santos, user302797, user26857, Christopher, Gibbs Nov 22 at 11:07
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." – José Carlos Santos, user302797, user26857, Christopher, Gibbs
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
up vote
-1
down vote
favorite
up vote
-1
down vote
favorite
Show that if there is a function $g: B → A$, satisfying $g(f(x)) = x$ for all $x ∈ A$, and $f(g(x)) = x$ for all $x ∈ B$, then $f$ is a bijection, and inverse of $f = g$.
I'm not sure how to prove this.
algebra-precalculus
edited Nov 21 at 11:14
Robert Z
90.7k1057128
asked Nov 21 at 10:56
Patrick
335
Show that if there is a function $g: B → A$, satisfying $g(f(x)) = x$ for all $x ∈ A$, and $f(g(x)) = x$ for all $x ∈ B$, then $f$ is a bijection, and inverse of $f = g$.
I'm not sure how to prove this.
algebra-precalculus
algebra-precalculus
edited Nov 21 at 11:14
Robert Z
90.7k1057128
asked Nov 21 at 10:56
Patrick
335
edited Nov 21 at 11:14
Robert Z
90.7k1057128
asked Nov 21 at 10:56
Patrick
335
edited Nov 21 at 11:14
Robert Z
90.7k1057128
edited Nov 21 at 11:14
Robert Z
90.7k1057128
edited Nov 21 at 11:14
Robert Z
90.7k1057128
90.7k1057128
asked Nov 21 at 10:56
Patrick
335
asked Nov 21 at 10:56
Patrick
335
asked Nov 21 at 10:56
Patrick
335
335
closed as off-topic by José Carlos Santos, user302797, user26857, Christopher, Gibbs Nov 22 at 11:07
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." – José Carlos Santos, user302797, user26857, Christopher, Gibbs
If this question can be reworded to fit the rules in the help center, please edit the question.
closed as off-topic by José Carlos Santos, user302797, user26857, Christopher, Gibbs Nov 22 at 11:07
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." – José Carlos Santos, user302797, user26857, Christopher, Gibbs
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
up vote
0
down vote
Hints:
Equality $f(g(b))=b$ for all $bin B$ can be used to show surjectivity of $f$.
Equality $g(f(a))=a$ for all $ain A$ can be used to prove injectivity of $f$.
answered Nov 21 at 11:11
drhab
94.8k543125
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
Hints:
Equality $f(g(b))=b$ for all $bin B$ can be used to show surjectivity of $f$.
Equality $g(f(a))=a$ for all $ain A$ can be used to prove injectivity of $f$.
answered Nov 21 at 11:11
drhab
94.8k543125
add a comment |
up vote
0
down vote
Hints:
Equality $f(g(b))=b$ for all $bin B$ can be used to show surjectivity of $f$.
Equality $g(f(a))=a$ for all $ain A$ can be used to prove injectivity of $f$.
answered Nov 21 at 11:11
drhab
94.8k543125
add a comment |
up vote
0
down vote
up vote
0
down vote
Hints:
Equality $f(g(b))=b$ for all $bin B$ can be used to show surjectivity of $f$.
Equality $g(f(a))=a$ for all $ain A$ can be used to prove injectivity of $f$.
answered Nov 21 at 11:11
drhab
94.8k543125
Hints:
Equality $f(g(b))=b$ for all $bin B$ can be used to show surjectivity of $f$.
Equality $g(f(a))=a$ for all $ain A$ can be used to prove injectivity of $f$.
answered Nov 21 at 11:11
drhab
94.8k543125
answered Nov 21 at 11:11
drhab
94.8k543125
answered Nov 21 at 11:11
drhab
94.8k543125
answered Nov 21 at 11:11
drhab
94.8k543125
94.8k543125
add a comment |
add a comment |
