# Latihan Tentang Ordinal 3

Soal-soal berikut ini adalah tentang aritmetika ordinal, yaitu bermain-main operasi jumlah, kali, perpangkatan dengan bilangan-bilangan ordinal.

Latihan 11. Periksa pernyataan berikut:

1. $\omega + 1 = \omega'$
2. $1 + \omega = \omega + 1$
3. $1 + 3 = 3 + 1$.

Latihan 12. Periksa pernyataan berikut:

1. $1 \times 3 = 3 \times 1$
2. $2 \times \omega = \omega + \omega$
3. $\omega \times 2 = \omega + \omega$

Latihan 13. Periksa pernyataan berikut:

1. $(\omega + 2) + \omega < (\omega + \omega) + 3$
2. $(\omega \times 2) + \omega < (\omega \times \omega) + 2$
3. $(\omega \times \omega) < \omega \times (2 \times \omega)$.

Latihan 14. Apakah berlaku $\omega \times (\omega + \omega) = (\omega \times \omega) + (\omega \times \omega)$.

Latihan 11.

1. $\omega + 1 = (\omega + 0)' = \omega'$. Benar.
2. $1 + \omega = \{ x: x \in 1+n \} = \omega \}$. Salah.
3. $1 + 3 = (1+2)' = (1+1)''=(1+0)'''=1'''=0''''$.
$3 + 1 = (3+0)' = 3' = 0''''$.
Benar.

Latihan 12.

1. $1 \times 3 = (1 \times 2) + 1 = (1 \times 2)' = ((1 \times 1) + 1)' = (1 \times 1)'' = ((1 \times 0) + 1)''$ $= (1 \times 0)''' = 0'''$.
$3 \times 1 = (3 \times 0) + 3 = 0 + 3 = (0 + 2)' = (0 + 1)'' = (0+0)''' = 0'''$.
Benar.
2. $2 \times \omega = \{ x: x \in 2 \times n \} = \omega$.
$\omega \ne \omega + \omega$.
Salah.
3. $\omega \times 2 = (\omega \times 1) + \omega = ((\omega \times 0) + \omega) + \omega = (0 + \omega) + \omega$ di mana $0 + \omega = \{ x : x \in 0+n \} = \omega$. Jadi, $\omega \times 2 = \omega + \omega$. Benar.

Latihan 13.

1. $(\omega + 2) + \omega = \omega + (2 + \omega) = \omega + \omega < (\omega + \omega) + 3$. Benar.
2. $(\omega \times 2) + \omega < (\omega \times 3) < (\omega \times \omega) < (\omega \times \omega)+3$. Benar.
3. $\displaystyle \omega \times (2 \times \omega) = \omega \times \bigcup_{\beta < \omega} 2 \times \beta = \omega \times \bigcup_{n=0}^\infty 2 \times n = \omega \times \omega$. Salah.

Latihan 14.

$\omega \times (\omega + \omega) = \omega \times (\omega \times 2) = (\omega \times \omega) \times 2 = (\omega \times \omega) + (\omega \times \omega)$. Perhatikan bukti ini menggunakan sifat asosiatif dari perkalian.