# 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 LawsÂ  (1)

$p \wedge T \equiv p$

$p \vee F \equiv p$
pTp and T
010
111
pFp or T
000
101

Domination Laws (2)

$p \vee T \equiv T$

$p \wedge F \equiv F$

pTp or T
011
111
pFp and F
000
100

Idempotent Laws (3)

$p \vee p \equiv p$

$p \wedge p \equiv p$

ppp or p
000
111
ppp and p
000
111

Double Negation Law (4)

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

pnot pnot (not p)
010
101

Commutative Laws (5)

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

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

pqp or qq or p
0000
0111
1011
1111
pqp and qq and p
0000
0100
1000
1111

Associative Laws (6)

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

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

pqrp or qq or r(p or q) or rp or (q or r)
0000000
0010111
0101111
0111111
1001011
1011111
1101111
1111111

Distributive Laws (7)

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

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

pqrq and rp or (q and r)p or qp or r(p or q) and (p or r)
00000000
00100010
01000100
01111111
10001111
10101111
11001111
11111111

De Morgan’s Laws (8)

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

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

pqnot pnot qnot(p or q)not p and not q
001111
011000
100100
110000
pqnot pnot qnot(p and q)not p or not q
001100
011011
100111
110011

Absorption Laws (9)

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

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

pqp and qp or (p and q)
0000
0100
1001
1111
pqp or qp and (p or q)
0000
0110
1011
1111

Negation Laws (10)

$p \vee \neg p \equiv T$

$p \wedge \neg p \equiv F$

pnot pp or not p
01T
10T
pnot pp and not p
01F
10F