Nombor asli

Set bagi seluruh nombor asli ditandakan dengan ${\displaystyle \mathbb {N} }$ ;

${\displaystyle \mathbb {N} =\{1,2,3,4,5,6,7,\cdots \ \}}$ 

Kardinaliti bagi set nombor asli, iaitu ${\displaystyle n(\mathbb {N} )}$  ditandakan dengan aleph-null, ${\displaystyle \aleph _{0}}$ .

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Nombor asli mempunyai beberapa sifat-sifat yang boleh diperhatikan.

Jika ${\displaystyle a_{1}+a_{2}=a_{3}}$ , dimana ${\displaystyle a_{1}}$  dan ${\displaystyle a_{2}}$  ialah nombor asli, maka ${\displaystyle a_{3}}$  juga adalah nombor asli. Untuk persamaan${\displaystyle a_{1}-a_{2}=a_{3}}$ , jika ${\displaystyle a_{1}>a_{2}}$  maka ${\displaystyle a_{3}}$  masih nombor asli, jika ${\displaystyle a_{1}<a_{2}}$ , maka ${\displaystyle a_{3}}$  akan menjadi integer negatif dan jika ${\displaystyle a_{1}=a_{2}}$ , maka ${\displaystyle a_{3}=0}$ . Pernyataan tersebut boleh ditulis dalam pernyataan logik;

• ${\displaystyle \forall (a_{1},a_{2}\in \mathbb {N} ),\,a_{1}+a_{2}=a_{3}\,\Rightarrow a_{3}\in \mathbb {N} }$ 
• ${\displaystyle \forall (a_{1},a_{2}\in \mathbb {N} ),\,\mathrm {jika} \,\,a_{1}>a_{2},\,a_{1}-a_{2}=a_{3}\,\Rightarrow a_{3}\in \mathbb {N} }$ 
• ${\displaystyle \forall (a_{1},a_{2}\in \mathbb {N} ),\,\mathrm {jika} \,\,a_{1}<a_{2},\,a_{1}-a_{2}=a_{3}\,\Rightarrow a_{3}\in \mathbb {Z} ^{-}}$ 
• ${\displaystyle \forall (a_{1},a_{2}\in \mathbb {N} ),\,\mathrm {jika} \,\,a_{1}=a_{2},\,a_{1}-a_{2}=a_{3}\,\Rightarrow a_{3}=0}$ 

Jika ${\displaystyle a_{1}\times a_{2}=a_{3}}$  dimana ${\displaystyle a_{1}}$  dan ${\displaystyle a_{2}}$  ialah nombor asli, maka ${\displaystyle a_{3}}$  juga adalah nombor asli.

• ${\displaystyle \forall (a_{1},a_{2}\in \mathbb {N} ),\,a_{1}\times a_{2}=a_{3}\,\Rightarrow a_{3}\in \mathbb {N} }$ 

Jika ${\displaystyle {\frac {a_{1}}{a_{2}}}=a_{3}}$  dimana ${\displaystyle a_{1}}$  dan ${\displaystyle a_{2}}$  ialah nombor asli, kalau ${\displaystyle a_{1}|\,a_{2}}$ , maka ${\displaystyle a_{3}}$  adalah nombor asli. Jika ${\displaystyle a_{1}\not \mid a_{2}}$ , maka ${\displaystyle a_{3}}$  ialah nombor nisbah positif.

• ${\displaystyle \forall (a_{1},a_{2}\in \mathbb {N} ),\,\mathrm {jika} \,\,a_{1}|\,a_{2}{\frac {a_{1}}{a_{2}}}=a_{3}\,\Rightarrow a_{3}\in \mathbb {N} }$ 
• ${\displaystyle \forall (a_{1},a_{2}\in \mathbb {N} ),\,\mathrm {jika} \,\,a_{1}\not \mid a_{2}{\frac {a_{1}}{a_{2}}}=a_{3}\,\Rightarrow a_{3}\in \mathbb {Q} ^{+}}$ 

bruh