99年私立銘傳大學資工所資傳所離散數學

1.

Prove that  is irrational

2.

Let G be the set of all nonzero real numbers and let a ∗ b = (ab)/2. Show that (G, ∗) is an Abelian group.

Let A = {x | x is an integer and x² < 16}. Identify each of the following as true or false.

3.

{1, 3, 5} ⊆ A

Let A = {x | x is an integer and x² < 16}. Identify each of the following as true or false.

4.

{−3, −2, −1} ⊆ A

Let A = {x | x is an integer and x² < 16}. Identify each of the following as true or false.

5.

{ } ⊆ A

Let A = {x | x is an integer and x² < 16}. Identify each of the following as true or false.

6.

{x | x is an integer and |x| < 5} ⊆ A

Let A = {x | x is an integer and x² < 16}. Identify each of the following as true or false.

7.

A ⊆ {−3, −2, −1, 0, 1, 2, 3}

Let S = (1 × 2 matrices, ？), where [x, y] ？ [w, z] = [x + w, (y + z)/2]. Determine which of the following properties hold for this structure

8.

Closure

Let S = (1 × 2 matrices, ？), where [x, y] ？ [w, z] = [x + w, (y + z)/2]. Determine which of the following properties hold for this structure

9.

Associative

10.

Find an explicit formula for the sequence defined by c_n = 3c_n−1 − 2c_n−2 with initial conditions c₁ = 5 and c₂ = 3