Tugas4 contoh pembuktian

: Dyaisa Algustavia Sekar Wido Laksono
: 4101417181
Dosen Pengampu : DR. Rochmad, M.SI.
Carilah beberapa contoh (minimal 2 contoh) pembuktian dalam matematika. Untuk setiap contoh
berilah ulasan secukupnya bagaimana alur berpikir dan logika dalam pembuktian tersebut.
1. Jumlah dari dua bilangan genap adalah bilangan genap
Penyelesaian :
Misalnya ada bilangan genap sembarang m dan n
Maka m = 2k dengan k adalah suatu bilangan bulat
Maka n = 2i dengan I adalah suatu bilangan bulat
m + n = 2k + 2i
= 2(k + i), (k + i) juga bilangan bulat
Jadi m + dapat ditulis 2 kali suatu bilangan bulat.
Sesuai define bilangan genap, maka m + n bilangan genap juga
2. Jika 𝑛2 bilangan ganjil, maka n bilangan ganjil
Bukti :
Untuk membuktikan pernyataan tersebut kita akan membuktikan kebenaran
𝑝 = 𝑛2 bilangan ganjil
𝑞 = 𝑛 bilangan ganjil
Jika p → q benar maka akan dibuktikan ~p → ~q juga benar.
Andaikan n bukan bilangan ganjil
Maka n bilangan genap
Sehingga n dinyatakan dengan n = 2k dengan k bilangan asli
Akibatnya 𝑛2 = (2𝑘)2 = 4𝑘 2 = 2(2𝑘)2
Artinya 𝑛2 bilangan genap
Jadi pengandaian bahwa n bukan bilangan ganjil adalah BENAR
Sehingga kontraposisi ~p → ~q juga benar.
Jadi implikasi p → q benar yang berarti 𝑛2 bilangan ganjil maka n adalah bilangan ganjil.
3. Buktikan bahwa jumlah dua bilangan ganjil adalah bilangan genap.
Dimisalkan m dan n dengan m dan n adalah ganjil
Jika m dan n ganjil
Maka terdapat bilang bulat p dan q
Sehingga m = 2p +1 dan n = 2q + 1
m + n = (2p + 1) + (2q + 1)
= 2p + 2q + 2
= 2(p + q +1)
= 2k
Karena m + n = 2k untuk bilangan bulat k = p + q +1
Maka m + n merupakan bilangan genap
berdasarkan definisi bilangan genap.

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