Conjunction and disjunction


LEMMA    \(P \wedge (Q \vee R) \vdash P \wedge Q \vee P \wedge R\)

Proof available

LEMMA    \(P \wedge Q \vee P \wedge R \vdash P \wedge (Q \vee R)\)

Proof available

THEOREM   

Conjunction is left-distributive over disjunction.

$$P \wedge (Q \vee R) \dashv\vdash P \wedge Q \vee P \wedge R$$

Proof available

LEMMA    \((Q \vee R) \wedge P \vdash Q \wedge P \vee R \wedge P\)

Proof available

LEMMA    \(Q \wedge P \vee R \wedge P \vdash (Q \vee R) \wedge P\)

Proof available

THEOREM   

Conjunction is right-distributive over disjunction.

$$(Q \vee R) \wedge P \dashv\vdash Q \wedge P \vee R \wedge P$$

Proof available

LEMMA    \(P \vee Q \wedge R \vdash (P \vee Q) \wedge (P \vee R)\)

Proof available

LEMMA    \((P \vee Q) \wedge (P \vee R) \vdash P \vee Q \wedge R\)

Proof available

THEOREM   

Disjunction is left-distributive over conjunction.

$$P \vee Q \wedge R \dashv\vdash (P \vee Q) \wedge (P \vee R)$$

Proof available

LEMMA    \(Q \wedge R \vee P \vdash (Q \vee P) \wedge (R \vee P)\)

Proof available

LEMMA    \((Q \vee P) \wedge (R \vee P) \vdash Q \wedge R \vee P\)

Proof available

THEOREM    Disjunction is right-distributive over conjunction.

Proof available

THEOREM   

Conjunction absorbs disjunction.

$$P \wedge (P \vee Q) \dashv\vdash P$$

Proof available

THEOREM   

Disjunction absorbs conjunction.

$$P \vee (P \wedge Q) \dashv\vdash P$$

Proof available