nasin pali pi jan Pu

nasin pali pi jan Pu (toki Inli: Boolean algebra) li kipisi sona pi nanpa nimi lon sona nanpa. ni la, nanpa pi nanpa nimi li lon (o sitelen e 1) anu ala (o sitelen e 0). ni la, pali nanpa li pali lon tan ni: pali pi wan lon (o sitelen e ∧), pali pi tu lon (o sitelen e ∨) en pali pi ala lon (o sitelen e ¬).

jan So Pu li pana e nasin pali lon lipu The Mathematical Analysis of Logic (lon tenpo sike 1847) lon lipu An Investigation of the Laws of Thought (lon tenpo sike 1854). nasin pali li suli lon pali pi ilo wawa, lon toki ilo. ona li kepeken lon sona pi kulupu ijo, lon sona pi pana nanpa.

nanpa

nasin pali pi jan Pu la, sitelen pi toki nanpa li sitelen e nanpa lon tan ni: ala en lon. nanpa ni li sitelen e 0 e 1. 0 en 1 li suli sona Pi. ona li ala nanpa pi wan nanpa, taso ona li lon nasin nanpa ma pi ijo tu GF(2). nasin nanpa ma GF(2) li sama nasin nanpa awen pi kipisi tu ni: $1+1=0$. ni^ la, pali pi wan nanpa li sama pali XOR (nanpa lon li ante la, o lon taso). ni^ la, pali pi mute nanpa li sama pali AND (lon en lon la, o lon taso). pali pi tu lon $x\lor y$ li kon la, $x+y-xy$. pali pi ala lon $\neg x$ li kon la, $1-x$. lon nasin nanpa ma GF(2) la, - li ken ante e + tan ni: ona li sama.

pali

lukin sona: lipu lon

pali wan

nasin pali pi jan Pu li jo e pali ni:

• pali pi wan lon (pali li ijo tu la, o sitelen e AND ($\land$))
• pali pi tu lon (pali li ijo tu la, o sitelen e OR ($\lor$))
• pali pi ala lon (pali li ijo wan la, o sitelen e NOT ($\neg$))

ni^ li toki e pali pi jan Pu.

lon nasin pali, nanpa nimi li toki e nanpa nimi pi jan Pu. nanpa nimi ni li lon anu ala. pali pi nanpa wan li kepeken e nanpa nimi pi jan Pu la, ona li ni:

pali lon	pali	sitelen	sitelen ante	kon	toki pona
pali pi wan lon	AND	$x\land y$	$x\;{\textrm {AND}}\;y,\,{\textrm {K}}xy$	$x=y=1{\text{ la, }}x\land y=1;{\text{ ante la, }}x\land y=0.$	$x{\text{ en }}y{\text{ li lon}}$
pali pi tu lon	OR	$x\lor y$	$x\;{\textrm {OR}}\;y,\,{\textrm {A}}xy$	$x=y=0{\text{ la, }}x\lor y=0;{\text{ ante la, }}x\lor y=1.$	$x{\text{ anu }}y{\text{ li lon}}$
pali pi ala lon	NOT	$\neg x$	${\textrm {NOT}}\;x,\,{\textrm {N}}x,\,{\bar {x}},\,x',\,!x$	$x=1{\text{ la, }}\neg x=0{\text{; kin la, }}x=0{\text{ la, }}\neg x=1.$	$x{\text{ li ala}}$

ante la, nanpa pi $x\land y$ en nanpa pi $x\lor y$ en nanpa pi $\neg x$ li ken sitelen lon lipu lon e ni:

$x$	$y$	$x\land y$	$x\lor y$
0	0	0	0
1	0	0	1
0	1	0	1
1	1	1	1

$x$	$\neg x$
0	1
1	0

pali tu

pali ni li wan e pani wan. ona li ni:

pali pi ken lon:	${\textstyle x\rightarrow y=\neg {x}\vee y}$
pali pi sama lon:	${\textstyle x\leftrightarrow y=(x\land y)\lor (\neg x\land \neg y)=(x\lor \neg y)\land (\neg x\lor y)}$
pali pi ante lon (pali XOR):	${\textstyle x\oplus y=\neg (x\leftrightarrow y)=(x\vee y)\wedge \neg (x\wedge y)=(x\vee y)\wedge (\neg x\vee \neg y)=(x\wedge \neg y)\vee (\neg x\wedge y)}$

kon ni li pana e lipu lon ni:

$x$	$y$	$x\rightarrow y$	$x\oplus y$	$x\leftrightarrow y,x\equiv y$
0	0	1	0	1
1	0	0	1	0
0	1	1	1	0
1	1	1	0	1

pali pi ken lon

pali wan la, x → y, anu Cxy li toki e pali pi ken lon (toki Inli: material implication). x li lon la, x → y li sama y (x li lon la en y li ala la, x → y li ala). taso x li ala, x → y li ala.

pali pi ante lon (pali XOR)

pali tu la, x ⊕ y anu Jxy li toki e pali pi ante lon (toki Inli: exclusive or) anu pali XOR tan ante pali pi tu lon (pali OR). ona li weka e ni: x en y li lon (o lukin lipu ni^).

pali pi sama lon

pali tu wan la, x ≡ y anu Exy li pali pi sama lon (toki Inli: equivalence) anu ala pi pali XOR. x li sama y la, ni li lon.

lawa

lawa pi nasin pali li sama lon e toki pi jan Pu li awen ni: $x\lor (y\lor z)=(x\lor y)\lor z$. toki pi jan Pu li ni: toki li wan e nimi nanpa e nanpa lon 0 e nanpa lon 1 e pali ∧ e pali ∨ e pali ¬. kon ni li ken suli e pali ⊕ e pali → e pali ≡ taso ni: ni li ike.

lawa pi nasin wan

lawa pi nasin pali li sama lawa pi nasin noka nanpa tan ni: ∨ li sama pali pi wan nanpa en ∧ li sama pali pi mute nanpa. anpa la, lawa ni li sama.

kulupu ante pi pali $\lor$:		$x\vee (y\vee z)$	$=(x\vee y)\vee z$
kulupu ante pi pali $\land$:		$x\wedge (y\wedge z)$	$=(x\wedge y)\wedge z$
ante pi pali $\lor$:		$x\vee y$	$=y\vee x$
ante pi pali $\land$:		$x\wedge y$	$=y\wedge x$
pana pi pali $\land$ lon pali $\lor$:		$x\wedge (y\vee z)$	$=(x\wedge y)\vee (x\wedge z)$
awen pi pali $\lor$:		$x\vee 0$	$=x$
awen pi pali $\land$:		$x\wedge 1$	$=x$
weka pi pali $\land$:		$x\wedge 0$	$=0$

anpa la, lawa ni li sama ala.

weka pi pali $\lor$:		$x\vee 1$	$=1$
awen pi pali $\lor$:		$x\vee x$	$=x$
awen pi pali $\land$:		$x\wedge x$	$=x$
moku 1:		$x\wedge (x\vee y)$	$=x$
moku 2:		$x\vee (x\wedge y)$	$=x$
pana pi pali $\lor$ lon pali $\land$:		$x\vee (y\wedge z)$	$=(x\vee y)\wedge (x\vee z)$

nasin wan li ni: nanpa nimi li ante la, ale li awen anu li ante sama (nanpa nimi li ante tan 0 tawa 1 la, ale li ken ala pi ante tan 1 tawa 0). ni li lon la, pali li pali pi nasin wan.

lawa pi nasin mute

pali pi ala lon li kon e lawa tu ni.

${\begin{aligned}&{\text{kipisi wan 1}}&x\wedge \neg x&=0\\&{\text{kipisi wan 2}}&x\vee \neg x&=1\end{aligned}}$

pali pi ala lon la, lon ale tan lawa sewi.

${\begin{aligned}&{\text{ala lon tu}}&\neg {(\neg {x})}&=x\end{aligned}}$

nasin pali pi jan Pu li jo e lawa pi Te Mokan:

${\begin{aligned}&{\text{lawa pi Te Mogan 1}}&\neg x\wedge \neg y&=\neg {(x\vee y)}\\&{\text{lawa pi Te Mogan 2}}&\neg x\vee \neg y&=\neg {(x\wedge y)}\end{aligned}}$