QUESTION TWO (a) Explain the following terms as used in discrete structures (6 marks) (i) Contradictions (ii) Contingency (iii) Propositional equivalence (b) Express the following logical function to its standard Sum of Product (4 marks) A(B+CD)+BC' (c) Consider the following conditional statement: If the flood destroys my house or the fires destroy my house then my insurance company will pay me. Write the converse, inverse.and contrapositive of the statement. (4 marks) (d) Discuss reasoning rules used to prove if an argument form is valid. (6 marks)

(a) (i) Contradictions: In discrete structures, a contradiction refers to a statement that is always false, regardless of the truth values of its components. It is a logical inconsistency that arises when two or more statements cannot be true at the same time.(ii) Contingency: A contingency is a statement that is neither always true nor always false. It is a logical relationship between two statements where the truth of one statement depends on the truth of the other statement.(iii) Propositional equivalence: Propositional equivalence refers to the relationship between two statements that are logically equivalent, meaning that they have the same truth value in all possible scenarios. If one statement is true, the other statement is also true, and vice versa.(b) The logical function can be expressed in standard Sum of Product form as .(c) Converse: If my insurance company pays me, then the flood destroys my house or the fires destroy my house.Inverse: If the flood does not destroy my house and the fires do not destroy my house, then my insurance company will not pay me.Contrapositive: If my insurance company does not pay me, then the flood does not destroy my house and the fires do not destroy my house.(d) Reasoning rules used to prove if an argument form is valid include:- Modus Ponens: If P implies Q and P is true, then Q must be true.- Modus Tollens: If P implies Q and Q is false, then P must be false.- Disjunction Elimination: If either P or Q is true, and both P and Q are false, then a contradiction arises.- Conjunction Introduction: If both P and Q are true, then the conjunction P and Q is true.- Hypothetical Syllogism: If P implies Q and Q implies R, then P implies R.- Transitive Property: If P implies Q and Q implies R, then P implies R.