discrete math for conputer science

Using only the derivation rules in Tables 1.11 and 1.12. but not 1.14. on the accompanying sheet, write a proof sequence to prove that the argument ( QP) ( PQ). called contraposition, is valid. TABLE 1.11 Expression Pro PAO (PIVOVR (PAQ) AR (PV) (PAQY Equivalence Rules Equivalent to Name/Abbreviation for Rule QVP Commutative-comm QAP PVOVR) Associative-ass PA(QAR) ' De Morgan's Laws-De PVO Morgan prva Implication-imp ( Double negation.dn ( PQ) A( OP) Definition of equivalence-equ Q. P PO TABLE 1.12 Inference Rules Can Derive o P From P.P. PQ, P. PN Name/Abbreviation for Rule Modus ponens-mp Modus tollens-mt Conjunction-con Simplification--sim Addition-add PAQ Pa PVQ TABLE 1.14 From P-00-R PVO. P: PO 0-P P PVP POR P.PL PAIOVA PVO More Inference Rules Can Derive Name/Abbreviation for Rule P-R [Example 16] Hypothetical syllogism-hs Q Exercise 25) Disjunctive syllogism-ds QP[Exercise 26 Contraposition-cont P Q Exercise 271 Contraposition-cont PAPExercise 28) Self-reference-self P (Exercise 201 Self-reference -- self P + 10 + Exercise 30) Exportation exp QlExercise 311 Inconsistency in PAVIP Exercise 321 Distributive-dist PVO) RE 33 Distributive-dist