Tugas Anreal

  • June 2020
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Nama : 1. Dian Novitasari (107017000745) 2. Nur Malitasari (107017000734) 3. Zulkahfi TEOREMA 1.2.1 Misalkan S ⊂ N dan S ≠ 0, maka S memiliki unsur terkecil, yaitu terdapat nο ∈ S, sehingga nο ≤ n, ∀n ∈ S. Contoh: 1. misal : S ={1,2,3,4,5} no = 1 no ∈ S no ≤ n 2. misal : S ={2,5,8,11} no = 2 no ∈ S no ≤ n 3. misal : S ={5,10,15} no = 5 no ∈ S no ≤ n 4. misal : S ={7,14,21} no = 7 no ∈ S no ≤ n 5. misal : S ={9,10,11,12} no = 9 no ∈ S no ≤ n LEMMA 1.2.2 Jika x, y ∈ Q dan x < y, maka terdapat z ∈ Q, sehingga x < z < y. Contoh : 1. misal : x =

2 5

y=

2 4

3 3 3 3 6 3 → 7 6,5 6 7 13 6

x

x
z y

3. misal : x =

5 7

5 6

y=

x
maka terdapat z 5 5 5 5 10 5 → 7 6,5 6 7 13 6

x
x z y

4. misal : x =

3 8

y=

3 7

x
maka terdapat z 3 3 3 →3 6 3 8 7,5 7 8 15 7

x
x z y

5. misal : x =

1 8

y=

1 7

x
maka terdapat z 1 8

1 7,5

x

z

1 1 2 → 7 8 15

1 7

x
y

x
maka terdapat z 2 2 5 4,5

x

z

2 2 → 4 5

y

2. misal : x =

3 7

x
maka terdapat z

4 9

2 4

x
y=

3 6

LEMMA 1.2.3 (Sifat Archimedes) Jika x ∈ Q, maka terdapat n ∈ z shg x < n. Contoh : 1. misal : x =

16 maka terdapat n = 4 5

x
16 <4 5 6 2. misal : x = maka terdapat n = 1 7

x
6 <1 7

23 3. misal : x = maka terdapat n = 6 4

x
23 <6 4 9 4. misal : x = maka terdapat n = 3 4

x
9 <3 4 8 5. misal : x = maka terdapat n = 2 7

x
8 <2 7

TEOREMA 1.4.1 (Sifat Archimedes) Untuk setiap x, y, ∈ R dan x > 0 terdapat n ∈ N, sehingga n x > y. Contoh : 1. misal : x = 4 , y = 2 terdapat n = 1 n.x >y 1.4 >2 4 >2 2. misal : x = 11 , y = 10 terdapat n = 2 n.x >y 2 . 11 > 10 22 > 10 3. misal : x = 9 , y = 16 terdapat n = 3 n.x >y 3 . 9 > 16 27 > 16 4. misal : x = 15 , y = 25 terdapat n = 4 n.x >y 4 . 15 > 25 60 > 25 5. misal : x = 8 , y = 51 terdapat n = 8 n.x >y 8 . 8 > 51 64 > 51

TEOREMA 1.4.2 Untuk setiap x, y ∈ R dan x < y terdapat p ∈ Q sehingga x < p < y Contoh : 1. misal : x = 5 , y = 6 x
11 2

6

x
26 3

13

x
76 < 26 3

x
−12 < −2 5

x
−13 < −5 2

x 0 dan n ∈ N terdapat x ∈ R sehingga xn = a. Contoh : 1. misal : a = 8 , n = 4 xn = a x = n a = 4 8 = 1,681792831 x∈R 2. misal : a = 25 , n = 7

xn = a x = n a = 7 25 = 1,583819609 x∈R 3. misal : a = 17 , n = 2 xn = a x = n a = 2 17 = 4,123105626 x∈R 4. misal : a = 200 , n = 3 xn = a x = n a = 3 200 = 5,848035476 x∈R 5. misal : a = 1151 , n = 6 xn = a x = n a = 6 1151 = 3,237272221 x∈R

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