Guardian Angel - Cinta Laura

This is my favorite song..

I like this song so much..

Enjoy it!

Guardian Angel - Cinta Laura

bahagianya ku punyamu
berharganya bersamamu
selalu kan ku jaga semuanya ini
semoga kan abadi

reff:
akhirnya ku temukan
you’re my guardian angel
ku mohon selamanya
seindah ini

akhirnya ku miliki
you’re my guardian angel
terjawab segalanya
kau yang ku nanti
baby i love you
(baby i love you) 

love you

bahagianya ada kamu
berharganya cinta kamu
selalu kan ku jaga semuanya ini
semoga kan abadi

repeat reff

you’re my angel

repeat ref

Takbiran

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Exponents and Logarithms

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A Rabbit and Twenty Crocodiles

Once upon a time, a rabbit wanted to cross a river but he could not swim. He had an idea. He saw a boss of crocodile swimming in the river. The rabbit asked the boss of crocodile, “How many crocodiles are there in the river?” The boss of crocodile answered, “We are twenty here.”

“Where are they?” the rabbit asked for the second time. “What is it for?” the boss of crocodile asked.

“All of you are good, nice, gentle and kind, so I want to make a line in order. Later I will know how kind you are,” said the rabbit. Then, the boss of crocodile called all his friends and asked them to make a line in order from one side to the other side of the river. Just then, the rabbit started to count while jumping from one crocodile to another; one…two…three…four… until twenty, and finally he thanked all crocodiles because he had cross the river.

                                                                                                         Nikmatul Maula / XII IPA 2 / 23

The Golden Touch

There once lived a king named Midas who was very rich. He loved two things above all else: his little daughter and his gold. The king had many rooms full of gold, but he wasn’t satisfied. He wanted to prossess even more gold.

One day when the king was in his garden, an unknown visitor came to him. He told the king that he could wish for anything he wanted. At first, the king didn’t believe the stranger, but finally he was convinced. “I wish to have more gold,” the king said. “Very well.” Answered the stranger. “You shall have your wish. Beginning tomorrow, everything you touch will turn to gold.”

The next morning the king got out of bed early. He wanted to see if his wish had been granted. First, he touched a chair, then a flower, then a table. Everything he touched turned into gold! The king was very happy. Then he sat down to eat his breakfast. But as soon as he touched his food, it turned to gold. The king began to worry. What would happen if he couldn’t eat anything?

Just then the little princess came into the dining room. She ran to her father. But as he touched her, she turned to gold.

The king was sad and worried. He went out to the garden. As he was walking, he saw the stranger again. The king begged him to take back the wish. “I don’t want any more gold,” he cried. The stranger warned the king to be absolutely sure this time. The he agreed to take back the wish.

From that day on, the king was a wiser and happier man, even though he had less gold.

                                  Nikmatul Maula / XII IPA 2 / 23

Calculation Tools

People need calculation tools to speed and ease the calculation. People need it because they often calculate. Calculation tools not only make people easy to calculate, but also make calculation process faster and increase the accuration of calculation results.

A long history of calculation, people have used many calculation tools. At the beginning, calculation tools is very simple; the finger, the list, barrier, pellet, etc. Beside it, people use the ruler, plate, etc. Afterward, calculation tools get the development with discovering calculator.

The capability of  calculation tools are different. Calculate with finger or toe is difficult, but with different tools, it is easier and practical. For example, calculate something using table or list, we don’t need to do the manual calculation, but we just see the answer in the table.

The various of calculation tools are the finger, the list of calculation, barrier, pellet, ruler, calculator, computer, etc.

Since people have known calculating, people have felt the important of calculation tools to make the calculation easier. Due to the progress of life, calculation tools get the development, from simple tools to modern tools, like calculator and computer.

Pengantar Dasar Matematika

**Task 1**

4.a) 1. [(a ∨ c) ∧ ~b] → [(d→ c) → f]
        2. ~a→ b
        3. ~b                                               /∴ [(d→ c) → f]
        4. a (2,3 MT)
        5. a ∨ c (4 add)
        6. (a ∨ c) ∧ ~b (5,3 konj)
        7. (d→ c) → f (1,6 MP)

4.b) 1. e→ (f ∧~g)
        2. (f ∨ g)→ h
        3. e                                                 /∴ h
        4. (f ∧~g)  (1,3 MP)
        5. f (4 simp)
        6. (f ∨ g) (5 add)
        7. h (2,6 MP)

4.c) 1. e→ f
        2. e→ g     /∴ e→ (f ∧ g)
        3. (e→ f) ∧ (e→ g)  (1,2 konj)
        4. e→ (f ∧ g) (3 dist)

4.d) 1. (~u ∨ v) ∧ (u ∨ v)
        2. ~x→ ~w                         /∴ v ∨ x
        3. (~u ∧ u) ∨ v          (1 dist)
        4. F ∨ v                   (3 komp)
        5. v          (4 id)
        6. v ∨ x (5 add)

4.e) 1. e→ f
        2. g→ f                           /∴(e ∨ g) → f
        3. ~f→ ~e                 (1 ekiv)
        4. ~f→ ~g                 (2 ekiv)
        5. (~f→ ~e) ∧ (~f→ ~g) (3,4 konj)
        6. ~f→ (~e ∧ ~g) (5 dist)
        7. (e ∨ g) → f (6 ekiv)

5.a) 1. b → n
        2. ~b → s               / ∴ n ∨ s
        3. ~n → ~b (1 ekiv)
        4. ~n → s (3,2 sil)
        5. n ∨ s         (4 ekiv)

     b.) 1. ( p ∧ t) → n
           2. ( t → n) → s
           3. p                            / ∴ s
           4. p → (t → n)  (1 eksp)
           5. t → n           (4,3 MP)
           6. s        (2,5 MP)

      d.) 1. b ∨ k
            2. ( b ∨ m )  → ( l ∧ h )
            3.  ~l                        / ∴ k
            4. ( ~l ∨ ~h)  → ( ~b ∧ ~m)    (2 ekiv)
            5. ~l ∨ ~h             (3 add)
            6. ~b ∧ ~m            (4,5 MP)
            7. ~b      (6 simp)
            8. ~b → k       (1 ekiv)
            9. k           (8,9 MP)

**Task 2**

Show that :
a) A ∩ A = A
b) A ∩ B = B ∩ A
c) ( A ∩ B ) ∩ C = A ∩ ( B ∩ C )

Answer

a) Proof :

i. Show that A ∩ A ⊂ A
Take any x ∈ A ∩ A
Obvious x ∈ A ∩ A
≡ x ∈ A ∧ x ∈ A
≡ x ∈ A (idempoten)
So, A ∩ A ⊂ A .................................(1)

ii. Show that A ⊂ A ∩ A
Take any x ∈ A
Obvious x ∈ A
≡ x ∈ A
≡ x ∈ A ∧ x ∈ A (idempoten)
So, A ⊂ A  ∩ A .................................(2)

From (1) and (2), we conclude that A ∩ A = A


b) Proof :

i. Show that ( A ∩ B ) ⊂ ( B ∩ A )
Take any x ∈ A ∩ B
Obvious x ∈ A ∩ B
≡ x ∈ A ∧ x ∈ B
≡ x ∈ B ∧ x ∈ A (komutatif)
So, ( A ∩ B ) ⊂ ( B ∩ A )......................(1)

ii. Show that( B ∩ A ) ⊂ ( A ∩ B )
Take any x ∈ B ∩ A
Obvious x ∈ B ∩ A
≡ x ∈ B ∧ x ∈ A
≡ x ∈ A ∧ x ∈ B (komutatif)
So, ( B ∩ A ) ⊂ ( A ∩ B ) ......................(2)

From (1) and (2), we conclude that A ∩ B = B ∩ A

c) Proof :

i. Show that ( A ∩ B ) ∩ C = A ∩ ( B ∩ C )
Take any x ∈ ( A ∩ B ) ∩ C
Obvious x ∈ ( A ∩ B ) ∩ C
≡ (x ∈ A ∧ x ∈ B) ∧ x ∈ C
≡ x ∈ A ∧ ( x ∈ B ∧ x ∈ C ) (asosiatif)
So, [( A ∩ B ) ∩ C] ⊂ [A ∩ ( B ∩ C )]...........(1)

ii. Show that A ∩ ( B ∩ C )  = (A ∩ B ) ∩ C
Take any x ∈  A ∩ ( B ∩ C )
Obvious x ∈ A ∩ ( B ∩ C )
≡ x ∈ A ∧ ( x ∈ B ∧ x ∈ C)
≡ ( x ∈ A ∧  x ∈ B ) ∧ x ∈ C  (asosiatif)
So, [A ∩ ( B ∩ C )] ⊂ [( A ∩ B ) ∩ C] ...........(2)

From (1) and (2), we conclude that ( A ∩ B ) ∩ C = A ∩ ( B ∩ C )

Mars Himatika UNNES

MARS HIMATIKA

Himpunan mahasiswa matematika

Membimbin mahasiswa kreatif berdaya guna

Bertaqwa pada Tuhan Yang Maha Esa

Setia pada nusa dan bangsa

Tunjukkan semangat generasi muda

Untuk membangun bangsa Indonesia

Berdedikasi tinggi dan penuh motivasi

Mengabdi pada ibu pertiwi

Himatika……Himatika……

Tunjukkanlah kiprahmu

Himatika……Himatika……

Jayalah selalu

Himatika……Himatika……

Jayalah Himatika kita

Jayalah Himatika kita

ORDINARY DIFFERENSIAL EQUATION

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Pascal Programming

1. Find the factor of a number
2. Calculate the sum of sequene
3. Count the deviation standard

download this file.

STRING in Pascal Programming

Maybe you already know what is a string variable from previous program examples that you have tried, but you don't know what operations and what functions one can apply to them and how can they be manipulated in order to obtain another form of string whatsoever.

In this lesson we will cover some important functions that the Pascal programming language has for us in order to cater for string operations far more easier to use than you think.

A.      Introduction

String in Pascal Language can be operated in many pupose. Some of string can be arranged and can be used as a condition that has been selected. Pascal provide some of standard procedure and function of string operation.

B.       Combination of String

String only has one operator, it’s ‘+’. The length of string that permit by Pascal is 255 characters. If this operation use to add the numeric, its function is adding two elements of numerical value. But if this operation use in string, then its function is combine two elements  of string value.

for more completely, download this paper.

GARIS DAN BIDANG YANG SEJAJAR

Definisi 5.1

Garis yang bersilangan adalah dua garis yang tidak berpotongan dan tidak terletak pada bidang yang sama.

Definisi 5.2

Sebuah garis dan bidang adalah sejajar, jika tidak mempunyai titik persekutuan.

Definisi 5.3

Bidang yang sejajar adalah bidang yang tidak mempunyai titik persekutuan.

Definisi 5.4

Sebuah garis melintang adalah garis yang memotong dua  garis yang sebidang di dua titik yang berbeda.

selengkapnya dapat didownload di sini: http://www.mediafire.com/?mhfb7l67fv3a7u9

Probabilities

it's about:

• Rules of Filling the Provided Place
• Factorial
• Permutation
• Permutation with some identical elements
• Cyclical permutation
• Permutation with some identical elements
• Combination
• Newton’s Binomial
• Expectation of an Event’s Frequency
• Probability of The Complementary of an Event
• Probability of mutually exclusive events
• Probability of the two events
• Probability of Mutually Independent Events
• Probability of the Conditional Events

and some examples and challenging problems.

please download here: http://www.mediafire.com/?mdo590hazo3kd5b

Geometri Netral

GEOMETRI NETRAL

      PENDAHULUAN

Dalam Geometri Euclid, Geometri Netral tidak menggunakan postulat ke-5 Euclid ataupun ingkaran dari postulat ke-5 itu. Aksioma ke-5 Euclides (kesejajaran) berbunyi ”Jika dua garis dipotong oleh sebuah garis transversal sedemikian hingga membuat jumlah sudut dalam sepihak kurang dari 180°, maka kedua garis itu berpotongan pada pihak yang jumlah sudut dalam sepihaknya kurang dari 180°. Aksioma ini diubah oleh Playfair dalam kalimat yang berbeda tetapi bermakna sama yaitu: ”Hanya ada satu garis yang sejajar dengan garis yang diketahui yang melalui sebuah titik di luar garis yang tidak diketahui.”

Dari kelima aksioma Euclides, jika aksioma kesejajaran dihilangkan maka geometri ini dinamakan geometri netral. Geometri netral ini menggunakan teorema-teorema Saccheri tanpa aksioma kesejajaran (Saccheri menganut postulat kesejajaran Euclides).

Dengan melakukan modifikasi-modifikasi, banyak proposisi dalam geometri netral adalah benar    secara  geometri  Euclid  maupun  non  Euclid.  Sebagai  akibatnya,  Geometri  Netral menyiapkan    kerangka  kerja  yang  cocok  yang  dengannya  kita  dapat  membandingkan  dan mempertentangkan sifat-sifat geometri Euclid dan non Euclid. Geometri netral akan menjelaskan peran postulat kesejajaran dalam geometri Euclid, membukakan jalan untuk mempelajari geometri non Euclid pada bab berikutnya, dan menghasilkan teorema yang cocok untuk geometri non Euclid.

Hal lain yang mendasar dalam geometri netral ini yaitu kemungkinan adanya persegi panjang atau kemungkinan tidak adanya persegi panjang. Jika pada geometri netral mengandung persegi panjang, maka jumlah besar sudut-sudut dalam setiap segitiga adalah 180°. Perlu diketahui juga bahwa pada geometri netral ada segi empat yang penting, yaitu yang dinamakan segi empat Saccheri. Sedangkan pada geometri Euclides, tidak ada perbedaan antara segiempat Saccheri dengan persegi panjang.

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Konvers, Invers, dan Kontraposisi

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