Exercise 5.9. Suppose P and R are atomic propositions. Show that P ⇒ Ris equivalent to its contrapositive and that P ⇒ R is not equivalent to itsconverse. For the latter, illustrate with an example using a language bychoosing P and R such that P ⇒ R is true and its converse is not.

The contrapositive of p→r is ~r→~p. so, the truth table will be, p r ~p ~r p→r ~r→~p T T F F T…

Answered: Exercise 5.9. Suppose P and R are…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Answer #3 using logical equivalence with the laws of logic
1. Rewrite the statements so that negations appear only within predicates (so that no negation is outside a quantifier or an expression involving logical connectives) (1) -Vx (D(x) (A(x) V M(x))) 2. Prove that the following expressions are logically equivalent by applying the laws of logic (1 (pq) (p V q) and T 3. Use a truth table to show whether the logical expression is a tautology, contradiction or neither -(p→q) → p
Construct direct proofs to show that the following symbolic arguments are valid. Commas mark the breaks between premises. ~C→(F→C), ~C ∴ ~F
1. Find the contrapositive, converse and negation of the following statement. (Va € N) (Vb ≤ Z) (a² − b² > 0 ⇒ (a ≤ b) V (a + b ≤ N)) Contrapositive: • Converse: • Negation:
Consider the propositional language with only → and ⊥ as connectives. Is that lan- guage truth functionally complete?
Evaluate the proposition ∀∃y.P(x,y) ⇐⇒ ∃y.∀x.P(x,y) for an unknown predicate P. Could the statement be true? Must the statement be true for all P? Give a specific P (with domains for x and y) where it is either true or false. Prove that it is true for all P, or show a contradiction.
3. Let us consider the propositional language where: A = "Peter does his homework" B = "Burt does his homework" C = "Andy does his homework" D = "Dexter comes to the party" Write the statement for the following symbols. (d) C → (~D → ~A) (e) (~D → C) A (D → ~B) (f) A A~BAC → ~D) a ( A a B → (D → C)
We define the following propositions • m: I like mango juice • e: Mango juice is expensive • b: I will buy mango juice. Consider the sentence "If I don't like mango juice and it is expensive, I will not buy mango juice". Which statement is logically equivalent to the given sentence. O (mv -e) → b O -b v (-m A e) -b A (-m A e) O b→ (m v ¬e)

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Exercise 5.9. Suppose P and R are atomic propositions. Show that P⇒ R is equivalent to its contrapositive and that P = → R is not equivalent to its converse. For the latter, illustrate with an example using a language by choosing P and R such that P ⇒ R is true and its converse is not.