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.

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