Logical equivalence


Logical equivalence

Hello friends, Welcome to my blog mathstips4u.
In my last post we have seen double implication or bi-conditional and its truth table.
In this post we are going to learn logical equivalence and some of its examples.
First we shall see what is meant by Statement Pattern.
Let p, q, r, …be simple statements. Then a statement formed from these statements and one or more connectives Ʌ, V, ~, →, ↔ is called a statement pattern.
e.g.  (i) p Ʌ   ̴q (ii) p Ʌ (p V q) (iii) p Ʌ (q ↔ r) etc. are statement patterns.
Now we shall see Logical equivalence.
Two statement patterns say  and  are said to logically equivalent if they have identical truth values  in their last column of the truth tables.
In that case we write  ≡   or  =
Ex. Using truth table verify
1. ~ (p V q) ≡ ~ p Ʌ ~ q      
2. ~ (p Ʌ q) ≡~ p V ~q
I shall verify first, the second example is left for you as an exercise.
The results (1) and (2) are called as De Morgan’s Laws
3.  Hence p → q ≡   ~p V q ≡   ~ q → ~p.    We shall see the truth table.  
p
q
p → q
̴ p
̴ q
̴p V q
̴ q →   ̴ p
T
T
T
F
F
T
T
T
F
F
F
T
F
F
F
T
T
T
F
T
T
F
F
T
T
T
T
T
(1)
(2)
(3)
(4)
(5)
(6)
(7)
We observed that column no’s (3), (6) and (7) are identical.
Hence p → q ≡   ̴p V q ≡   ̴ q →   ̴ p  
So    ~q → ~p is contrapositive of p → q. 
4. p ↔ q ≡ (p → q) Ʌ (q → p). We shall see the truth table.  
p
q
p ↔ q
p → q
a
q → p
b
a Ʌb
T
T
T
T
T
T
T
F
F
F
T
F
F
T
F
T
F
F
F
F
T
T
T
T
(1)
(2)
(3)
(4)
(5)
(6)

We observed that column no. (3) and column no (6) are identical
 Hence p ↔ q ≡ (p → q) Ʌ (q → p)
Ex. Using truth table verify that
1. p Ʌ (q V r) ≡ (p Ʌ q) V (p Ʌ r)
We shall see the truth table. 
p
q
r
q V r
p Ʌ (q V r)
p Ʌ q
= a
p Ʌ r
= b
a V b
T
T
T
T
T
T
T
T
T
T
F
T
T
T
F
T
T
F
T
T
T
F
T
T
F
T
T
T
F
F
F
F
T
F
F
F
F
F
F
F
F
T
F
T
F
F
F
F
F
F
T
T
F
F
F
F
F
F
F
F
F
F
F
F
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
We observed that column no. (5) and column no, (8) are identical
Hence p Ʌ (q V r) ≡ (p Ʌ q) V (p Ʌ r)
2. p V (q Ʌ r) ≡ (p V q) Ʌ (p V r)
This example is left for you as an exercise.
These results are called Distributive laws.
In this way we have seen statement pattern and Logical equivalence.
In my next post we will learn converse, Inverse and contrapositive of an implication.

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