(a) Define what it means for a set of propositions to be consistent.Show that the set {a, (¬b), (a → b)} is inconsistent. (b) Find a set T = {p1,Y2,

Y3, P4} consisting of four propositions so that I is inconsistent and any proper subset of I is consistent.Give a natural deduction proof that I is inconsistent, illustrate this proof with a Venn diagram involving the sets V (y;), and also prove that {varphi1, varphi2, varphi3} is consistent. (c) Give natural deduction proofs of the following: \text { (i) }\{\alpha \rightarrow(\neg \beta \rightarrow \gamma), \alpha \rightarrow(\neg \beta \rightarrow \neg \gamma)\} \vdash \alpha \rightarrow \beta \text { (ii) }\{(\alpha \rightarrow \neg \beta) \rightarrow(\gamma \rightarrow \alpha), \neg \beta, \gamma \rightarrow(\alpha \rightarrow \delta)\} \vdash \gamma \rightarrow \delta