臺大數學系新生暑假作業 2022/08 請書寫完整，於

臺大數學系新生暑假作業 2022/08

請書寫完整，於 08/31 22:00 前繳交至 Google 表單https://forms.gle/

r8XVZ5YXhU74gTik8，參加 09/02 新生日將使用並檢討本份作業。

1. (a) If a is a rational number and b an irrational number, show that a + b is an irrational number. What about ab?

(b) Show that √ⁿ

2 is not a rational number for any integer n≥ 2.

(c) Show that log₁₀2 is not a rational number.

2. Find all integer solutions (x, y)∈ Z² such that 2022x + 678y = 18. (Hint: Use Euclidean algorithm.)

3. On I = [a, b]⊂ R, consider the statement P : “There exists x ∈ I such that for every y∈ I, x ⩽ y”. Which of the following statements are ¬P ?

(a) For every y∈ I, there exists x ∈ I such that x ⩽ y.

(b) There exists y∈ I such that for every x ∈ I, x ⩽ y.

(c) For every x∈ I, there exists y ∈ I such that x ⩽ y.

(d) There exists x∈ I such that for every y ∈ I, x > y.

(e) There exist x∈ I and y ∈ I such that x > y.

(f) For every x∈ I and y ∈ I, x > y.

(g) For every y∈ I, there exists x ∈ I such that x > y.

(h) For every x∈ I, there exists y ∈ I such that x > y.

(i) There exists y∈ I such that for every x ∈ I, x > y.

4. Determine if the following statements are true or false. Justify your answers.

(a) ∀ x ∈ R, ∃ y ∈ R such that y²= 4x.

(b) ∃ y ∈ R such that ∀ x ∈ R we have y²= 4x.

(c) ∀ y ∈ R, ∃ x ∈ R such that y²= 4x.

(d) ∃ x ∈ R such that ∀ y ∈ R we have y²= 4x.

(e) For every subset A⊆ N, there exists a ∈ A such that for every x ∈ A, a ⩽ x.

(f) There is a subset A⊆ N such that for every a ∈ A, there exists x ∈ A such that a < x.

(g) There is a subset A⊆ N such that for every a ∈ A, there exists x ∈ A such that x < a.

5. Let X be a set and A, B, C⊆ X. Prove the following statements:

(a) A∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) and A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C).

(b) A\ (B ∪ C) = (A \ B) ∩ (A \ C) and A \ (B ∩ C) = (A \ B) ∪ (A \ C).

(c) A∩ B = ∅ if and only if A ⊆ B^c.

6. The following problems deal with some relations between a mapping and opera- tions on sets. Given a function f : A→ B, we have two definitions:

• The image of a subset X⊆ A under the function f is the set:

f (X) ={f(x) | x ∈ X}.

• The preimage of a subset Y ⊆ B under the function f is the set:

f⁻¹(Y ) ={x ∈ A | f(x) ∈ Y }.

Prove the following statements:

(a) For any subset X ⊆ A, we have X ⊆ f⁻¹(f (X)). Also, give an example to explain that the equality may not hold (namely, X⊊ f⁻¹(f (X)) is possible).

(b) For any subset Y ⊆ B, we have f(f⁻¹(Y )) ⊆ Y . Also, give an example to explain that the equality may not hold (namely, f (f⁻¹(Y ))⊊ Y is possible).

(c) Assume that f is injective. Then, for any subset X⊆ A, we have f(A\X) ⊆ B\ f(X). Does this statement hold if f is not assumed to be injective?

(d) For any subsets Y1, Y₂⊆ B, we have f⁻¹(Y₁)\ f⁻¹(Y₂) = f⁻¹(Y₁\ Y2).

(e) For a collection{Xi}ⁿi=1 of subsets of A with Xi⊆ A, we have:

i. f ( _n

∪

i=1

X_i )

=

∪n i=1

f (X_i)

ii. f ( _n

∩

i=1

X_i )

⊆

∩n i=1

f (X_i). Also, give an example to explain that the

equality may not hold (namely, f ( _n

∩

i=1

Xi

)

⊊

∩n i=1

f (Xi) is possible).

(f) For a collection{Yj}^mj=1 of subsets of B with Yj ⊆ B, we have:

i. f⁻¹



∪^m

j=1

Yj



 = ∪^m

j=1

f⁻¹(Yj).

ii. f⁻¹



∩^m

j=1

Yj



 = ∩^m

j=1

f⁻¹(Yj).

7. Let f : A→ B and g : B → C be maps.

(a) Show that if both f and g are injections, then so is g◦ f.

(b) Show that if both f and g are surjections, then so is g◦ f.

(c) Show that if both f and g are bijections, then so is g◦ f.

(d) Show that if g◦f is an injection, then f is also an injection. Give an example where g is not an injection but g◦ f is.

(e) Show that if g◦f is a surjection, then g is also a surjection. Give an example where f is not a surjection but g◦ f is.

(f) If h : B→ A is a map such that h ◦ f = 1Aand f◦ h = 1B, show that both f and h are bijections.

8. The unit sphere S inR³ is the set S ={(x, y, z) ∈ R³| x²+ y²+ z² = 1}. Let N = (0, 0, 1)∈ S and M = S \ {N}. Consider the xy-plane L = {(X, Y, 0) | X, Y ∈ R}. The stereographic projection f : M → L is a mapping which sends a point P ∈ M to a point P^′ ∈ L where P^′ is the unique intersection point of the line through N and P with the planeL .

(a) Derive the stereographic projection formula:

f (x, y, z) =

( x

1− z, y 1− z, 0

) .

(b) Show that f is a bijection and find its inverse explicitly.

L

P^′(X, Y, 0) P (x, y, z)

N

Figure 1: The stereographic projection

9. Fix n∈ N, and for any a, b ∈ Z, let the relation a ∼ b mean n | (a − b).

(a) Show that ∼ is an equivalence relation on Z.

Remark: The equivalence class of a will be denoted by ¯a or ¯an, and the set of these equivalence classes will be denoted by Z/nZ := {¯k : k ∈ Z}. This set has n elements.

(b) Check that if a∼ b, c ∼ d, then a + c ∼ b + d.

(c) Check that if a∼ b, c ∼ d, then ac ∼ bd.

Here, we will explain “well definedness” for the next exercise. This is an error hard to find when you are writing a proof.

You may want to define the following “taking the numerator” function:

f :Q → Z : f(r) = p when r = p q.

This looks like a function but it is actually not a function, for a rational number r = ^p_q may have another expression (say ^2p_2q), which makes r map to multiple values. In this situation, we say f is not well-defined.¹

Let us see another example:

g :Z/10Z → Z/10Z : f(¯a) = a².

Although ¯a can be represented by many a (e.g. a + 10), such a² are all the same equivalence class (by exercise 9(c)). In this situation, we say g is well-defined. That is, g maps every element in the domain to exactly one element.

The problem arises only when the element in the domain has several representations (like r = ^p_q = ^kp_kq) and we use these representations to define our function. For example, in defining a function f :Q → Q, there is no danger in f(r) = 1 + r², but we have to be careful with f (r) = q²+ p²

q² for r = p q.

When we say a “function”²f is well-defined, it means that f is indeed a function.

That is, f maps every element in the domain to exactly one element.³

In mathematics, we check a function f : X → Y is well-defined by showing the following: If α, β represent the same element in X, then f (α) = f (β).

1You may ask p, q to be coprime and q > 0 in order to define this function well, but this is off-topic.

2Before the well definedness is checked, f is not yet a function!

3You can also treat addition as a function (with two variables). In the previous exercise, we checked that addition and multiplication onZ/nZ is well-defined.

10. (a) Show that f :Q → Q: f(r) = ^p2_q^+q2 ² for r = ^p_q, is well-defined.

(b) Show that f :Z/10Z → Z/5Z: f(¯a) = ¯a is well-defined. (Hint: These two ¯a are different!)

(c) Show that f :Z/10Z → Z/3Z: f(¯a) = ¯a is not well-defined.

(d) Is f :Z/10Z → Z/20Z: f(¯a) = a² well-defined? Justify your answer.

11. Prove the following identities:

(a) sin α cos β =sin(α + β) + sin(α− β)

2 , sin α + sin β = 2 sinα + β

2 cosα− β 2 . (b) cos α sin β =sin(α + β)− sin(α − β)

2 , sin α− sin β = 2 cosα + β

2 sinα− β 2 . (c) cos α cos β = cos(α + β) + cos(α− β)

2 , cos α + cos β = 2 cosα + β

2 cosα− β 2 . (d) sin α sin β =−cos(α + β)− cos(α − β)

2 , cos α− cos β = −2 sinα + β

2 sinα− β 2 . (e) 1

2+ cos x + cos 2x + cos 3x +· · · + cos nx = sin[(

n +¹₂) x] 2 sin^x₂

12. (a) Find a number δ > 0 such that|x²− 4| < 1

2 if|x − 2| < δ.

(b) Find a number δ > 0 such that |√

x²+ 5− 2| < 1 if |x − 1| < δ.

(You have to justify your answer.)

13. Use the mathematical induction to prove the following statements.

(a) Prove that for all integers n≥ 4, we have 3ⁿ> n³. (b) Define a sequence{cn}^∞n=0as follows:

{

c_n+1= ⁴⁹₈c_n−²²⁵₈ c_n−2, n≥ 2, c0= 0, c1= 2, c2= 16.

Prove that cn= 5ⁿ− 3ⁿ for all n∈ N ∪ {0}.

(c) Suppose that A = (

7 12

−2 −3 )

. Prove that for any n∈ Z,

Aⁿ =

(−2 −6

1 3

) + 3ⁿ

(

3 6

−1 −2 )

.

Here A⁻ⁿ= (Aⁿ)⁻¹ is the inverse of Aⁿ, and we follow the convention that A⁰=

( 1 0 0 1 )

is the identity matrix.

14. (a) Show that for any matrix A = (

a b c d

)

∈ M2×2(R), we have the following relation:

A²− tr(A) · A + det(A) · I2= O2

where I2= (

1 0 0 1 )

, O2= (

0 0 0 0 )

, tr(A) = a + d and det(A) = ad− bc.

(b) Let A = (

1 4 2 3 )

. Evaluate the matrix A⁵− 4A⁴− 3A³− 7A²− 16A − 2I2.

15. (Arithmetic and geometric means inequality)

(a) Show that for two non-negative real numbers a, b∈ R⩾0, we have a + b

2 ⩾√ ab

and that the equality holds if and only if a = b.

(b) Show that for any k∈ N, and 2^k non-negative real numbers a1, a₂, . . . , a₂k∈ R⩾0, we have

1 2^k

2^k

∑

i=1

a_i ≥ ^2k√a₁a₂· · · a2^k−1a₂k

and that the equality holds if and only if a1= a2=· · · = a2^k.

(c) Suppose it is true that for any n (n≥ 2) non-negative real numbers a1, . . . , an∈ R⩾0, we have

1 n

∑n i=1

ai⩾ √ⁿ

a1a2· · · an−1an

and that the equality holds if and only if a1 = a2 =· · · = an. Show that the inequality remains true for n− 1 non-negative real numbers and that the equality holds if and only if they are equal. (Hint: Consider an =

1

n−1(a1+· · · + an−1).)

(d) Show that for any n non-negative real numbers a1, . . . , an∈ R⩾0, we have 1

n

∑n i=1

ai⩾ √ⁿ

a1a2· · · an−1an

and that the equality holds if and only if a1= a2=· · · = an.

16. (Cauchy–Schwarz inequality) For real numbers a1, . . . , an, b1, . . . , bn ∈ R, show that

(a1b1+ a2b2+· · · + anbn)²⩽ (a²1+ a²₂+· · · + a²n)(b²₁+ b²₂+· · · + b²n) and that the equality holds if and only if there exists some constants c1, c2 ∈ R, which are not all 0, such that c1(a1, . . . , an) + c2(b1, . . . , bn) = (0, . . . , 0). (Hint:

The quadratic polynomial ∑ⁿ

i=1

(aix + bi)² in x is always non-negative.)

17. (Triangle inequality) Write ∥x∥ =

√∑n

k=1

x²_k for x = (x1, x₂, . . . , x_n)∈ Rⁿ. Show that

∥x + y∥ ⩽ ∥x∥ + ∥y∥

for all x, y∈ Rⁿ and that the equality holds if and only if there exist some non- negative constants c1, c2∈ R, which are not all 0, such that c1x = c2y.

(When n = 1, this inequality is just|a + b| ≤ |a| + |b| for all a, b ∈ R.)

提醒：如果不理解題目該證明什麼，請至新生講義裡確認定義後再作答，並多多查 詢英文單字字義。