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Mendelson: 'Introduction to mathematical logic' bases Propositional Calculus on the connectives 
and 
. Those two connectives form a complete set of connectives in the sense that any Boolean function of arity 1 or more can be expressed using nothing but function parameters and those two connectives. As an example, Boolean 'or' may be expressed thus:
There are, however, two Boolean functions of arity zero. They are the Boolean constants 
(truth) and 
(falsehood). One cannot define them from and without resorting to something extra like a free variable or an arbitrary constant.
To get a completely complete set of connectives, we use and instead of and . And we define and thus:
Having and allows to state Propositional Calculus differently from the way Mendelson does. But we nevertheless follow Mendelson closely so that you can sit with Mendelsons book and the present pages and see a close correspondence.
Having and , however, also allows to formulate and prove some new proofs. Here is a proof that is indeed true:
And here comes a proof which states that if we can prove 
under the assumption then we can drop the assumption:
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