Implication connective - Modus ponens rule:

We notice that this is a trivial example, so it highlights how we use truth tables: as the first line is the only one when both above-line propositions so A and A→B are be true. According to that we see this line the proposition B is also true. Where this shows us that there we have an entailment: as the above-line propositions entail the below-line one. 

To justify here why such inference rules are valuable so remember what the main application of automated deduction is use: to prove theorems. Hence theorems are normally part of a larger theory has axioms. There axioms are very special theorems that are taken tobe true without question. Thus whenever we have a theorem statement we want to prove than we should be able to began from the axioms and deduce the theorem statement requiring sound inference rules as modus ponens.