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
pTp and T
010
111
pF p 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
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