#12 part c using material implication and logical equivilancy Use De Morgan\'s l
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#12 part c using material implication and logical equivilancy
Use De Morgan's laws to find the negation of each of the following statements. a) Kwame will take a job in industry or go to graduate school b) Yoshiko knows Java and calculus. c) James is young and strong. d) Rita will move to Oregon or Washington. Show that each of these conditional statements is a tautology by using truth tables. a) (p q) rightarrow p b) p rightarrow (p q) c) p rightarrow (p rightarrow q) d) (p q) rightarrow (p rightarrow q) e) (p rightarrow q) rightarrow p f) (p rightarrow q) rightarrow q Show that each of these conditional statements is a tautology by using truth tables. a) [p (p q)] rightarrow q b) [(p rightarrow q) (q rightarrow r)] rightarrow (p rightarrow r) c) [p (p rightarrow q)] rightarrow q d) [(p q) (p rightarrow r) (q rightarrow r)] rightarrow r Show that each conditional statement in Exercise 9 is a tautology without using truth tables. Show that each conditional statement in Exercise 10 is a tautology without using truth tables. Use truth tables to verify the absorption laws. a) p (p q) p b) p (p q) p Determine whether (p (p rightarrow q)) rightarrow is a tautology. Show that (p rightarrowq) (q rightarrow r) rightarrow (p rightarrow r) is a tautology. Show that (p q) (p r) rightarrow (q r) is a tautology. Show that (p rightarrow q) rightarrow r and p rightarrow (q rightarrow r) are not logically equivalent. Show that (p rightarrow q) rightarrow (r rightarrow s) and (p rightarrow r) rightarrow (q rightarrow s) are not logically equivalent. The dual of a compound preposition that contains only the logical operations and is the compound preposition obtained by replacing each by each by each T by F, and each F by T. The dual of s is denoted by s*. Find the dual of each of these compound prepositions. a) p q b) p (q (r T)) c) (p q) (q F) Find the dual of each of these compound prepositions. a) p q r b) (p q r) s c) (p F) (q T) When does s* = s, where s is a compound proposition? Show that (s*)* = s when s is a compound proposition. Show that the logical equivalences in Table 6, except for the double negation law, come in pairs, where each pair contains compound propositions that are duals of each other.Explanation / Answer
The following proof uses simplification. You will find another below it that does not
(p ^ (p -> q)) -> q
=> (p ^ (~p v q)) -> q Material implication
=> ((p ^ ~p) v (p ^ q)) -> q Distributive law
=> (F v (p ^ q) -> q
=> (p ^ q) -> q
=> q -> q Simplification
Note: If you do not understand simplification, (a ^ b) -> a or (a ^ b) -> b is simplification
Without simplification, the proof uses disjunctive syllogism.
(p ^ (p -> q)) -> q
=> (p ^ (~p v q)) -> q Material implication
=> ((~p v q) ^ p) -> q Commutative law
=> q -> q Disjunctive syllogism
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