Question

Let   f:   A→B   be   bijective.       Prove   that   for   each   b   in   B,   there   exists   a&

Let   f:   A→B   be   bijective.       Prove   that   for   each   b   in   B,   there   exists   a   unique   a   in   A   such   that   f(a)   =   b.

Math   Advanced-Math   
0 0
Add a commentTranscribed image text
Next >

Homework Answers

Answer #1

by definition of bijectivity ,f is bijective implies f is is injective and surjective.by definition subjectivity for all b in B there exist an a in A such that f(a)=b. hence only we need to prove the uniqueness of such a in A.for this we suppose there exist two a1,a2 in A such that f(a1)=f(a2)=b. but since f is injective by definition of injectivity f(a1)=f(a2) implies a1=a2. So such a is unique.hence the proof.

0 0
Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Log In

Sign Up

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
Let f : A → B and suppose that there exists a function g : B...
Let f : A → B and suppose that there exists a function g : B → A such that (g ◦ f)(a) = a and (f ◦ g)(b) = b. Prove that g = f −1 . Thank you!
8.4: Let f : X → Y and g : Y→ Z be maps. Prove that...
8.4: Let f : X → Y and g : Y→ Z be maps. Prove that if composition g o f is surjective then g is surjective. 8.5: Let f : X → Y and g : Y→ Z be bijections. Prove that if composition g o f is bijective then f is bijective. 8.6: Let f : X → Y and g : Y→ Z be maps. Prove that if composition g o f is bijective then f is...
Let a < b, a, b, ∈ R, and let f : [a, b] → R...
Let a < b, a, b, ∈ R, and let f : [a, b] → R be continuous such that f is twice differentiable on (a, b), meaning f is differentiable on (a, b), and f' is also differentiable on (a, b). Suppose further that there exists c ∈ (a, b) such that f(a) > f(c) and f(c) < f(b). prove that there exists x ∈ (a, b) such that f'(x)=0. then prove there exists z ∈ (a, b) such...
Let f : [1, 2] → [1, 2] be a continuous function. Prove that there exists...
Let f : [1, 2] → [1, 2] be a continuous function. Prove that there exists a point c ∈ [1, 2] such that f(c) = c.
4) Let F be a finite field. Prove that there exists an integer n ≥ 1,...
4) Let F be a finite field. Prove that there exists an integer n ≥ 1, such that n.1F = 0F . Show further that the smallest positive integer with this property is a prime number.
A function f : A → B is called constant if there exists b ∈ B...
A function f : A → B is called constant if there exists b ∈ B such that for all x ∈ A, f(x) = b. Let f : A → B, and suppose that A is nonempty. Prove that f is constant if and only if for all g : A → A, f ◦ g = f.
Let f : A → B, and let V ⊆ B. (a) Prove that V ⊇...
Let f : A → B, and let V ⊆ B. (a) Prove that V ⊇ f(f−1(V )). (b) Give an explicit example where the two sides are not equal. (c) Prove that if f is onto then the two sides must be equal.
6. Let a < b and let f : [a, b] → R be continuous. (a)...
6. Let a < b and let f : [a, b] → R be continuous. (a) Prove that if there exists an x0 ∈ [a, b] for which f(x0) 6= 0, then Z b a |f(x)|dxL > 0. (b) Use (a) to conclude that if Z b a |f(x)|dx = 0, then f(x) := 0 for all x ∈ [a, b].
(a) Let a,b,c be elements of a field F. Prove that if a not= 0, then...
(a) Let a,b,c be elements of a field F. Prove that if a not= 0, then the equation ax+b=c has a unique solution. (b) If R is a commutative ring and x1,x2,...,xn are independent variables over R, prove that R[x σ(1),x σ (2),...,x σ (n)] is isomorphic to R[x1,x2,...,xn] for any permutation σ of the set {1,2,...,n}
Let a > 0 and f be continuous on [-a, a]. Suppose that f'(x) exists and...
Let a > 0 and f be continuous on [-a, a]. Suppose that f'(x) exists and f'(x)<= 1 for all x2 ㅌ (-a, a). If f(a) = a and f(-a) =-a. Show that f(0) = 0. Hint: Consider the two cases f(0) < 0 and f(0) > 0. Use mean value theorem to prove that these are impossible cases.
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT