### Logic and Mathematical Reasoning#### Exercise 5**a) Give the converse, the inverse, and the contrapositive of the following statement:** - **Original Statement:** "If a function \( f \) is differentiable, then the function \( f \) is continuous." - **Converse:** "If a function \( f \) is continuous, then the function \( f \) is differentiable." - **Inverse:** "If a function \( f \) is not differentiable, then the function \( f \) is not continuous." - **Contrapositive:** "If a function \( f \) is not continuous, then the function \( f \) is not differentiable."**b) Write each of the two statements in symbolic form and determine whether they are logically equivalent:** **Statements:** 1. "If you paid full price, you didn’t buy it at Crown Books." 2. "You didn’t buy it at Crown Books or you didn’t pay full price." **Symbolic Form:** - Let \( P \) represent "You paid full price." - Let \( Q \) represent "You bought it at Crown Books." - Statement 1: \( P \rightarrow \neg Q \) - Statement 2: \( \neg Q \vee \neg P \)**Analysis of Logical Equivalence:**- The structure of Statement 1, \( P \rightarrow \neg Q \), translates to "If P, then not Q."- The structure of Statement 2, \( \neg Q \vee \neg P \), translates to "Either not Q, or not P."Upon evaluating the logical structures, we identify that:- Statement 1: \( P \rightarrow \neg Q \) is logically equivalent to \( \neg P \vee \neg Q \), which is precisely the structure of Statement 2.**Conclusion:** The two given statements are logically equivalent.

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5) a) Give the converse, the inverse, and the contrapositive of the following: "If a function f is differentiable, then the function fis continuous". b) Write each of the two statements in symbolic form and determine whether they are logically equivalent : If you paid full price, you didn't buy it at Crown Books. You didn't buy it at Crown Books or you didn't pay full price.

5) a) Give the converse, the inverse, and the contrapositive of the following: "If a function f is differentiable, then the function fis continuous". b) Write each of the two statements in symbolic form and determine whether they are logically equivalent : If you paid full price, you didn't buy it at Crown Books. You didn't buy it at Crown Books or you didn't pay full price.

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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