清华微积分-1_Ch1习题

U3-1

Here are some sets:

(1) R both and

(2) ∅ both and

(3) (1,+∞) open set

(4) [−1,0]  closed set, -1 and 0 , which are not interior points, belong to the set.

(5) {1,2,3} none of them is interior point. They are isolated points. And the set is discrete set. closed set

(6) {y|y=2*x^2+1,x∈[0,2)} =[1,+∞)  closed set =[1,9), cuz 9 is a boundary point and is not included.

(7) Q × neither open set nor closed set

(8) Qc × neither open set nor closed set

Among the above sets, the total number of open set is: 3

the total number of closed set is: 5 4

 

U3-2

Consider the set S=[1,2)⋃{0}

Which of the following statements about S are TRUE?

x=1 is not an interior point of Sx=0 is not an interior point of Sx=0 is not a limit point of Sx=2 is not a limit point of S

Given a set S ⊂ R, a point l ∈ R is called a limit point £4Å:§ or point of accumulation(‡:) of the set S, if every deleted δ-neighborhood of l contains one or more points of S.

 

U4-2

Given the set of numbers S={1,1.1,0.9,1.01,0.99,1.001,0.999,...}

 

S={1}⋃{1+0.1n|n∈N}⋃{1−0.1n|n∈N}

∀a∈S,a≤1.1anda≥0.9

∀b<1.1,1.1∈S>b

∀c>0.9,0.9∈S<c

So 1.1 is the LUB of S, and 0.9 is the GLB of S.

∀ε>0,∃n∈N, s.t.1+0.1^n∈S and 1+0.1^n−1=0.1^n<ε  1的任意去心邻域和S的交集不为空

So 1 is a limit point of S.

 

U5-2

Given following numbers:

e, 

π, 

0,

(√3−√2)/(√3+√2),  = 5 - 2 √ 6  ==> x^2 - 10 x + 1 = 0

√2+√3+√5,  可构造出6次整数系数方程的解是√2+√3+√5

2+3i,  x^2 - 4 x +13 = 0

4/7

Of all the numbers above,

Which ones are algebraics? 

0, 4/7

Which ones are transcendentals? 

e, π, 

Which ones are irrational numbers?

e, π, (√3−√2)/(√3+√2), √2+√3+√5