Prove the following two statements: If the permutation α is even then α^-1 is even. If...

Prove the following two statements: If the permutation α is even then α^-1 is even. If the permutation α is odd then α^-1 is odd.

Expert Solution

Colby Messinger answered 4 months ago

prove that sign p=sign ^-p (if p is a permutation).

Prove the following statements! 1. If A and B are sets then (a) |A ∪ B|...

Prove the following statements! 1. If A and B are sets then (a) |A ∪ B| = |A| + |B| − |A ∩ B| and (b) |A × B| = |A||B|. 2. If the function f : A→B is (a) injective then |A| ≤ |B|. (b) surjective then |A| ≥ |B|. 3. For each part below, there is a function f : R→R that is (a) injective and surjective. (b) injective but not surjective. (c) surjective but not injective. (d)...

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....

Let α, β be cuts as defined by the following: 1) α ≠ ∅ and α...

Let α, β be cuts as defined by the following: 1) α ≠ ∅ and α ≠ Q 2) if r ∈ α and s ∈ Q satisfies s < r, then s ∈ α. 3) if r ∈ α, then there exists s ∈ Q with s > r and s ∈ α. Let α + β = {r + s | r ∈ α and s ∈ β}. Show that the set of all cuts R with the...

Prove the following statements! 1. There is a bijection from the positive odd numbers to the...

Prove the following statements! 1. There is a bijection from the positive odd numbers to the integers divisible by 3. 2. There is an injection f : Q→N. 3. If f : N→R is a function, then it is not surjective.

1. Determine if the following statements are true or false. If a statement is true, prove...

1. Determine if the following statements are true or false. If a statement is true, prove it in general, If a statement is false, provide a specific counterexample. Let V and W be finite-dimensional vector spaces over field F, and let φ: V → W be a linear transformation. A) If φ is injective, then dim(V) ≤ dim(W). B) If dim(V) ≤ dim(W), then φ is injective. C) If φ is surjective, then dim(V) ≥ dim(W). D) If dim(V) ≥...

1. a) Prove that if n is an odd number then 3n + 1is an even...

1. a) Prove that if n is an odd number then 3n + 1is an even number. Use direct proof. b) Prove that if n is an odd number then n^2+ 3 is divisible by 4. Use direct proof. 2. a) Prove that sum of an even number and an odd number is an odd number. Use direct proof. b) Prove that product of two rational numbers is a rational number. Use direct proof. 3. a) Prove that if n2is...

Prove that the following two statements are not logically equivalent. In your proof, completely justify your...

Prove that the following two statements are not logically equivalent. In your proof, completely justify your answer. (a) A real number is less than 1 only if its reciprocal is greater than 1. (b) Having a reciprocal greater than 1 is a sufficient condition for a real number to be less than 1. Proof: #2. Prove that the following is a valid argument: All real numbers have nonnegative squares. The number i has a negative square. Therefore, the...

Prove that if n is an integer and n^2 is even the n is even.

1) Construct a 2 × 2 examples to prove the following statements: (a) λ ∈ σ(A)...

1) Construct a 2 × 2 examples to prove the following statements: (a) λ ∈ σ(A) and µ ∈ σ(B) 6=⇒ λ + µ ∈ σ(A + B). (b) λ ∈ σ(A) and µ ∈ σ(B) 6=⇒ λµ ∈ σ(AB).

Question