A truth table is a convenient way of summarising the truth values of a statement. A truth table consists of rows and columns. The letter T is used to indicate the truth value of a true statement and the letter F is used to indicate the truth value of a false statement.
Truth nd Falsity of a conjunction
Example: Construct the truth table for p^q.
First, we have to list all the possible combinations of true and false for p and q.
There are 2 components p and q, there would be 2^{2} = 4 rows.
p  q  p ^ q 
T  T   
T  F   
F  T   
F  F   
Since we have agreed that p ^ q is true only when both p and q are true, the complete truth table is as shown below.
a. Let us consider the conjunction 5+3 = 8 and 71= 4. Here the component 5+3 = 8 is true but the component 71 = 4 is false. Hence, the conjunction is false.
b. '3 is an odd number and 4 is an even number ' is a true statement, since both the components
of this conjunction are true statements.
c. The statement 'A is hale and healthy' is true only if both the conditions are satisfied, i,e, A must be hale and A must be healthy. If either or both of these conditions is not satisfied, the conjunction is false statement.
Truth and Falsity of a disjunction
If the letters p and q represent two simple statements, the the truth value is as shown above.
Truth table for a negation (~p)
Since ~p consists of only one simple statement p, so there are 2^{1} = 2 rows in the truth table.
Example: Construct the truth table for the statement ~p v q.
 Step 1: Write in the first two columns the four possible pairs of truth values for the statements p and q.
p  q  ~p  ~p v q 
T  T     
T  F     
F  T     
F  F     
 Step 2: Using column 1 as reference, we negate the statement p.
p  q  ~p  ~p v q 
T  T  F   
T  F  F   
F  T  T   
F  F  T   
 Step 3: Using columns 3 and 2 and recalling that a disjunction is false only when both components (3 and 2) are false, we enter the values for ~p v q in the column labelled 4. We note that the statement ~p v q is false only in row 2, because in this case both components (~p, q) are false. The completed truth table is as shown below.
p  q  ~p  ~p v q 
T  T  F  T 
T  F  F  F 
F  T  T  T 
F  F  T  T 
Example: Construct the truth table for the statement ~(p^ q)
 Step 1: Write in the first two columns the four possible pairs of truth values for the statements p and q.
 Step 2: Write the values of ~q in column 3.
 Step 3: Recalling that a conjunction is true only when both its components are true, fill in the values or p ^ ~q in column 4. Now negate the values of p^~q and enter the values for ~(p^~q) in column 5.
p  q  ~q  (p^~q)  ~(p^~q) 
T  T  F  F  T 
F  F  T  T  F 
F  T  F  F  T 
F  F  T  F  T 
