# Preposition Logic and Problems – III

In this article , you will find common laws for preposition logic. These are very helpful in solving problems like simplifying a complex prepositional statement into a simple one. They also form the basis for arguments which you will learn in future lessons.

Identity Laws  (1)

$p \wedge T \equiv p$

$p \vee F \equiv p$
p T p and T
0 1 0
1 1 1
p F p or T
0 0 0
1 0 1

Domination Laws (2)

$p \vee T \equiv T$

$p \wedge F \equiv F$

p T p or T
0 1 1
1 1 1
p F p and F
0 0 0
1 0 0

Idempotent Laws (3)

$p \vee p \equiv p$

$p \wedge p \equiv p$

p p p or p
0 0 0
1 1 1
p p p and p
0 0 0
1 1 1

Double Negation Law (4)

$\neg(\neg p) \equiv p$

p not p not (not p)
0 1 0
1 0 1

Commutative Laws (5)

$p \vee q \equiv q \vee p$

$p \wedge q \equiv q \wedge p$

p q p or q q or p
0 0 0 0
0 1 1 1
1 0 1 1
1 1 1 1
p q p and q q and p
0 0 0 0
0 1 0 0
1 0 0 0
1 1 1 1

Associative Laws (6)

$(p \vee q) \vee r \equiv p \vee (q \vee r)$

$(p \vee q) \vee r \equiv p \vee (q \vee r)$

p q r p or q q or r (p or q) or r p or (q or r)
0 0 0 0 0 0 0
0 0 1 0 1 1 1
0 1 0 1 1 1 1
0 1 1 1 1 1 1
1 0 0 1 0 1 1
1 0 1 1 1 1 1
1 1 0 1 1 1 1
1 1 1 1 1 1 1

Distributive Laws (7)

$p \vee (q \wedge r) \equiv (p \vee q) \wedge (p \vee r)$

$p \wedge (q \vee r) \equiv (p \wedge q) \vee (p \wedge r)$

p q r q and r p or (q and r) p or q p or r (p or q) and (p or r)
0 0 0 0 0 0 0 0
0 0 1 0 0 0 1 0
0 1 0 0 0 1 0 0
0 1 1 1 1 1 1 1
1 0 0 0 1 1 1 1
1 0 1 0 1 1 1 1
1 1 0 0 1 1 1 1
1 1 1 1 1 1 1 1

De Morgan’s Laws (8)

$\neg( p \wedge q) \equiv \neg p \vee \neg q$

$\neg( p \vee q) \equiv \neg p \wedge \neg q$

p q not p not q not(p or q) not p and not q
0 0 1 1 1 1
0 1 1 0 0 0
1 0 0 1 0 0
1 1 0 0 0 0
p q not p not q not(p and q) not p or not q
0 0 1 1 0 0
0 1 1 0 1 1
1 0 0 1 1 1
1 1 0 0 1 1

Absorption Laws (9)

$p \vee (p \wedge q) \equiv p$

$p \wedge (p \vee q) \equiv p$

p q p and q p or (p and q)
0 0 0 0
0 1 0 0
1 0 0 1
1 1 1 1
p q p or q p and (p or q)
0 0 0 0
0 1 1 0
1 0 1 1
1 1 1 1

Negation Laws (10)

$p \vee \neg p \equiv T$

$p \wedge \neg p \equiv F$

p not p p or not p
0 1 T
1 0 T
p not p p and not p
0 1 F
1 0 F