# Conjunction

LEMMA    $$P \wedge P \vdash P$$

Proof available

LEMMA    $$P \vdash P \wedge P$$

Proof available

THEOREM

Conjunction is idempotent.

$$P \wedge P \dashv\vdash P$$

Proof available

THEOREM

Conjunction is commutative.

$$P \wedge Q \dashv\vdash Q \wedge P$$

Proof available

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

Proof available

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

Proof available

THEOREM

Conjunction is associative.

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

Proof available

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

Proof available

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

Proof available

LEMMA

Conjunction is left self-distributive.

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

Proof available

LEMMA

Conjunction is right self-distributive.

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

Proof available

THEOREM

Conjunction is self-distributive.

Proof available

LEMMA    $$\bot \wedge P \vdash \bot$$

Proof available

LEMMA    $$\bot \vdash \bot \wedge P$$

Proof available

THEOREM

False is a left zero element of conjunction.

$$\bot \wedge P \dashv\vdash \bot$$

Proof available

LEMMA    $$P \wedge \bot \vdash \bot$$

Proof available

LEMMA    $$\bot \vdash P \wedge \bot$$

Proof available

THEOREM

False is a right zero element of conjunction.

$$P \wedge \bot \dashv\vdash \bot$$

Proof available

LEMMA    $$\top \wedge P \vdash P$$

Proof available

LEMMA    $$P \vdash \top \wedge P$$

Proof available

THEOREM

True is a left identity element of conjunction.

$$\top \wedge P \dashv\vdash P$$

Proof available

LEMMA    $$P \wedge \top \vdash P$$

Proof available

LEMMA    $$P \vdash P \wedge \top$$

Proof available

THEOREM

True is a right identity element of conjunction.

$$P \wedge \top \dashv\vdash P$$

Proof available