Answered: Exercise 1. Express the following…

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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Question

Exercise 1.
Express the following statements symbolically using multiple quantifiers. De-
fine your domains and two-variable predicates. Then negate the statements.
(a) Every rational number when multiplied by some integer is an integer.
Domain(s):
• Predicate:
• Statement:
• Negation:
● Englsih translation of negation:
. Which is true? The original statement or its negation? Justify your
answer.

Expert Solution

Step 1: Discrete math quantifier problem

(a) Every rational number when multiplied by some integer is an integer.

Domain(s): $D colon space s e t space o f space r a t i o n a l space n u m b e r s comma straight rational numbers space a n d space E colon S e t space o f space i n t e g e r s comma straight integer numbers$

Predicate: $l e t comma P left parenthesis x comma n right parenthesis space r e p r e s e n t space x space i s space m u l t i p l i e d space b y space n space i s space i n t e g e r.$

Statement: $for all x space element of straight rational numbers space there exists space n element of straight integer numbers space P left parenthesis x comma n right parenthesis$

Negation: $not open square brackets for all x space element of straight rational numbers space there exists space n element of straight integer numbers space P left parenthesis x comma n right parenthesis close square brackets identical to there exists x element of straight rational numbers space for all n element of straight integer numbers space not P left parenthesis x comma n right parenthesis$

It mean that $" s o m e space r a t i o n a l space n u m b e r space w h e n space m u l t i p l i e d space b y space a n y space i n t e g e r space i s space n o t space i n t e g e r space "$