Preposition Logic and Problems

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	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