If p and q are mathematical statements, then in order to show that the statement “p and q” is true, the following steps are followed. 
Step-1 : Show that the statement p is true. 
Step-2 : Show that the statement q is true. 
    

  If p and q are mathematical statements, then in order to show that the statement “p or q” is true, one must consider the following.
Case 1 : By assuming that  p is false, show that q must be true. 
Case 2 : By assuming that q is false, show that p must be true. 
    

  In order to prove the statement “if p then q” we need to show that any one of the following case is true. 
Case 1 : By assuming that p is true, prove that q must be true.(Direct method) 
Case 2 : By assuming that q is false, prove that p must be false.(Contrapositive method) 
    

  In order to prove the statement “p if and only if q”, we need to show. 
(i)  If p is true, then q is true and   
(ii)  If q is true, then p is true