Prove the following statements! 1. Let S = {0, 1, . . . , 23} and...

Prove the following statements!

1. Let S = {0, 1, . . . , 23} and define f : Z→S by f(k) = r when 24|(k−r). If g : S→S is defined by

(a) g(m) = f(7m) then g is injective and

(b) g(m) = f(15m) then g is not injective.

2. Let f : A→B and g : B→C be injective. Then g ◦f : A→C is injective.

3. Let f : A→B and g : B→C be surjective. Then g ◦ f : A→C is surjective.

4. There is a surjection f : A→B such that f −1 : B→A is not a function.

Expert Solution

Colby Messinger answered 1 year ago

Question 1. Let F be an ordered ﬁeld. For each of the following statements, prove the...

Question 1. Let F be an ordered ﬁeld. For each of the following statements, prove the statement or provide a counterexample. (a) For all x,y,z,w ∈F, if x < y and xw < yz, then w < z. (b) If x,y,z,w ∈F, then |x + w|≤|x + y|+|y + z|+|z + w| Let x ∈R, a ∈R, and b ∈R. (a) Suppose that |x−a| = 3|x−b|. Let c =(9b−a)/ 8 . Prove that |x−c| = 3 8|a−b|

Let x, y ∈ R. Prove the following: (a) 0 < 1 (b) For all n...

Let x, y ∈ R. Prove the following: (a) 0 < 1 (b) For all n ∈ N, if 0 < x < y, then x^n < y^n. (c) |x · y| = |x| · |y|

Let Bt be a 1-dimensional Brownian motion and let c > 0 be a constant. Prove...

Let Bt be a 1-dimensional Brownian motion and let c > 0 be a constant. Prove that is also a Brownian motion.

Unless otherwise noted, all sets in this module are finite. Prove the following statements... 1. Let...

Unless otherwise noted, all sets in this module are finite. Prove the following statements... 1. Let S = {0, 1, . . . , 23} and define f : Z→S by f(k) = r when 24|(k−r). If g : S→S is defined by (a) g(m) = f(7m) then g is injective and (b) g(m) = f(15m) then g is not injective. 2. Let f : A→B and g : B→C be injective. Then g ◦f : A→C is injective. 3....

Let S ∈ R3 be the sphere of radius 1 centered on the origin. a) Prove...

Let S ∈ R3 be the sphere of radius 1 centered on the origin. a) Prove that there is at least one point of S at which the value of the x + y + z is the largest possible. b) Determine the point (s) whose existence was proved in the previous point, as well as the corresponding value of x + y + z.

Let p be an integer other than 0, ±1. (a) Prove that p is prime if...

Let p be an integer other than 0, ±1. (a) Prove that p is prime if and only if it has the property that whenever r and s are integers such that p = rs, then either r = ±1 or s = ±1. (b) Prove that p is prime if and only if it has the property that whenever b and c are integers such that p | bc, then either p | b or p | c.

Question

Prove the following statements! 1. Let S = {0, 1, . . . , 23} and...

Solutions

Expert Solution

Related Solutions

Question 1. Let F be an ordered ﬁeld. For each of the following statements, prove the...

Let x, y ∈ R. Prove the following: (a) 0 < 1 (b) For all n...

Let Bt be a 1-dimensional Brownian motion and let c > 0 be a constant. Prove...

Unless otherwise noted, all sets in this module are finite. Prove the following statements... 1. Let...

Let S ∈ R3 be the sphere of radius 1 centered on the origin. a) Prove...

Let p be an integer other than 0, ±1. (a) Prove that p is prime if...

. Let x, y ∈ R \ {0}. Prove that if x < x^(−1) < y...

Let (?,?,?) be a probability space and suppose that ?∈? is an event with ?(?)>0. Prove...

Let S ⊆ R and let G be an arbitrary isometry of R . Prove that...

2. Prove that |[0, 1]| = |(0, 1)|