(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) <br />(i) Contradictions: In discrete structures, a contradiction refers to a situation where two statements or propositions are in direct opposition to each other, and both cannot be true at the same time. For example, the statements "It is raining" and "It is not raining" are contradictory because they cannot both be true simultaneously.<br /><br />(ii) Contingency: A contingency in discrete structures refers to a statement or proposition that is neither a tautology (always true) nor a contradiction (always false). In other words, the truth value of a contingency depends on the specific circumstances or conditions under which it is evaluated. For example, the statement "It is raining" is a contingency because its truth value depends on whether it is actually raining or not.<br /><br />(iii) Propositional equivalence: Propositional equivalence in discrete structures refers to the relationship between two propositions that are logically equivalent, meaning that they have the same truth value in all possible scenarios. For example, the propositions "A and B" and "B and A" are propositionally equivalent because they have the same truth value for any given values of A and B.<br /><br />(b) The logical function $A(B+CD)+BC'$ can be expressed in standard Sum of Product (SOP) form as follows:<br />$A'B' + A'BC' + ABC + AB'CD$<br /><br />(c) <br />Converse: If my insurance company pays me, then the flood destroys my house or the fires destroy my house.<br />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.<br />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.<br /><br />(d) Reasoning rules used to prove if an argument form is valid include:<br />1. Modus Ponens: This rule states that if a conditional statement (if P then Q) is true and the antecedent (P) is true, then the consequent (Q) must also be true.<br />2. Modus Tollens: This rule states that if a conditional statement (if P then Q) is true and the consequent (Q) is false, then the antecedent (P) must also be false.<br />3. Disjunctive Syllogism: This rule states that if a disjunction (P or Q) is true and one of the disjuncts (P or Q) is false, then the other disjunct (P or Q) must be true.<br />4. Hypothetical Syllogism: This rule states that if a conditional statement (if P then Q) is true and another conditional statement (if Q then R) is true, then a new conditional statement (if P then R) must also be true.<br />5. Transitive Property: This rule states that if a conditional statement (if P then Q) is true and another conditional statement (if Q then R) is true, then a new conditional statement (if P then R) must also be true.