The Laws of Logic
The following results, known as Laws of logic, follows from the definition of logical equivalence.
In these laws T0 denotes a tautology and F0 denotes a contradiction.
1. Law of double negation
For any proposition p,   p   p
2. Idempotent Laws
For any proposition p,
(a)
 p  p  p
(b)  p  p   p
3. Identity Laws
For any proposition p,
(a)
 p  F0   p
4. Inverse laws
For any proposition p,
(a)
 p  p   T0
(b)  p  T0   p
(b)  p  p   F0
5. Domination Laws
For any proposition p,
(a)
 p  T0   T0
(b)  p  F0   F0
6. Commutative Laws
For any two proposition p and q,
(a)
 p  q  q  p
(b)  p  q    q  p 
7. Absorption Laws
For any two proposition p and q,
(a)  p   p  q   p
8. De Morgan’s Laws
For any two proposition p and q,
(a)   p  q   p  q
9. Associative Laws
(b)  p   p  q   p
(b)   p  q   p  q
For any three proposition p , q, r,
(b) p   q  r    p  q   r
(a) p   q  r    p  q   r
10. Distributive Laws
For any three proposition p , q, r,
(a) p   q  r    p  q    p  r  (b) p   q  r    p  q    p  r 
The above laws can be proved with aid of the truth tables.
Proof of 8(a): Let us prepare the truth table for   p  q  and p  q . The tables are shown
below in a combined form:
p
q
p
q
 p  q
 p  q 
p  q
0
0
1
1
0
1
1
0
1
1
0
0
1
1
1
0
0
1
0
1
1
1
1
0
0
1
0
0
From the above table, we note that   p  q  and p  q have the same truth values in all
possible situations. Hence   p  q   p  q .
Proof of 10(a): The truth values of p   q  r  and  p  q    p  r  for all possible truth values
of p, q, r, are shown in the following table:
p
q
r
q  r 
p  q  r 
0
0
0
0
1
1
1
1
0
0
1
1
0
0
1
1
0
1
0
1
0
1
0
1
0
0
0
1
0
0
0
1
0
0
0
1
1
1
1
1
 p  q  p  r   p  q   p  r 
0
0
1
1
1
1
1
1
0
1
0
1
1
1
1
1
0
0
0
1
1
1
1
1
From column 5 and 8 of the table, we find that p   q  r  and  p  q    p  r  have the same
truth values in all possible situations. Therefore, p   q  r    p  q    p  r  .
Note :

  p  q    p  q 

 p  q   p  q