In proposition logic, alphabets are used to represent atomic propositions, understood as a grammatically correct expression in formal language.
Every atomic proposition is either true or false and the combination of atomic propositions using logical connectives (such as and/or/if,then/iff) give rise to complex propositions.
For instance, the complex proposition (A & B) is true if and only if both atomic propositions A and B are true. In other instances, where either A or B or both are false, the complex proposition is false.
When trying to evaluate the truth conditions for complex propositions with many atomic propositions, one's required to compute the possible truth-value combinations among the different atomic propositions first. For instance, to evaluate the truth-conditions for the proposition (A&B), one needs to list the possible truth-value combinations as (A true B true, A false B False, A True B False, A False B True) first.
With this in mind, how do I prove, using mathematical induction, that the number of possible truth-value combinations for n propositions is 2^n?
I am not exposed to set theory and it is my humble request that any explanations involving set theory be as beginner-friendly as possible.