Prove 1. Let f : A→ B and g : B → C . If g...

Prove

1. Let f : A→ B and g : B → C . If g 。 f is one-to-one, then f is one-to-one.

2. Equivalence of sets is an equivalence relation (you may use other theorems without stating them for this one).

Expert Solution

Colby Messinger answered 1 year ago

4. Let a < b and f be monotone on [a, b]. Prove that f is...

4. Let a < b and f be monotone on [a, b]. Prove that f is Riemann integrable on [a, b].

Let F and G~be two vector fields in R2 . Prove that if F~ and G~...

Let F and G~be two vector fields in R2 . Prove that if F~ and G~ are both conservative, then F~ +G~ is also conservative. Note: Give a mathematical proof, not just an example.

(A) Let a,b,c∈Z. Prove that if gcd(a,b)=1 and a∣bc, then a∣c. (B) Let p ≥ 2....

(A) Let a,b,c∈Z. Prove that if gcd(a,b)=1 and a∣bc, then a∣c. (B) Let p ≥ 2. Prove that if 2p−1 is prime, then p must also be prime. (Abstract Algebra)

Let a < c < b, and let f be defined on [a,b]. Show that f...

Let a < c < b, and let f be defined on [a,b]. Show that f ∈ R[a,b] if and only if f ∈ R[a, c] and f ∈ R[c, b]. Moreover, Integral a,b f = integral a,c f + integral c,b f .

Prove the following: (a) Let A be a ring and B be a field. Let f...

Prove the following: (a) Let A be a ring and B be a field. Let f : A → B be a surjective homomorphism from A to B. Then ker(f) is a maximal ideal. (b) If A/J is a field, then J is a maximal ideal.

Let A, B be sets and f : A → B and g : B →...

Let A, B be sets and f : A → B and g : B → C . Characterize when g ◦ f : A → C is a bijection.

9. Let f be continuous on [a, b]. Prove that F(x) := sup f([x, b]) is...

9. Let f be continuous on [a, b]. Prove that F(x) := sup f([x, b]) is continuous on [a, b]

Let f: A→B and g:B→C be maps. (A) If f and g are both one-to-one functions,...

Let f: A→B and g:B→C be maps. (A) If f and g are both one-to-one functions, show that g∘f is one-to-one. (B) If g∘f is onto, show that g is onto. (C) If g∘f is one-to-one, show that f is one-to-one. (D) If g∘f is one-to-one and f is onto, show that g is one-to-one. (E) If g∘f is onto and g is one-to-one, show that f is onto. (Abstract Algebra)

(a) (f ∘ g)(3) (b) g(f(2)) (c) g(f(5)) (d) (f ∘ g)(−3) (e) (g ∘ f)(−1) (f) f(g(−1))

Let G be a group. For each x ∈ G and a,b ∈ Z+ a) prove...

Let G be a group. For each x ∈ G and a,b ∈ Z+ a) prove that xa+b = xaxb b) prove that (xa)-1 = x-a c) establish part a) for arbitrary integers a and b in Z (positive, negative or zero)

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