Please do not forget the justifications Consider the proposed proof of the statementProofStatement1. VA, B, CCU,C- (AUB) = (A - C) U (B-C)2.C-(AUB) = (AUB) - C= (AUB) NCC= (ANC) U (BNC)= (A-C) U (B-C)6...VA, B,CCU, C- (AUB) = (A - C) U (B-C) Transitivity ofequality.3.4.VA, B, CCU,C-(AUB) = (AC) U (B-C)5.JustificationTo be proved.Commutativity.Set differencelaw.Distributionlaw.Set differencelaw.a. Find a counterexample with three-empty sets that shows that the statement to be proved is false. Justify your counterexample.b. Find at least one statement in the proof that is not correct and create a counterexample with non-empty sets to prove your assertion. Justify your counterexample.

Answered: Consider the proposed proof of the…

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

Please do not forget the justifications

Consider the proposed proof of the statement
Proof
Statement
1. VA, B, CCU,C- (AUB) = (A - C) U (B-C)
2.
C-(AUB) = (AUB) - C
= (AUB) NCC
= (ANC) U (BNC)
= (A-C) U (B-C)
6...VA, B,CCU, C- (AUB) = (A - C) U (B-C) Transitivity of
equality.
3.
4.
VA, B, CCU,C-(AUB) = (AC) U (B-C)
5.
Justification
To be proved.
Commutativity.
Set difference
law.
Distribution
law.
Set difference
law.
a. Find a counterexample with three
-empty sets that shows that the statement to be proved is false. Justify your counterexample.
b. Find at least one statement in the proof that is not correct and create a counterexample with non-empty sets to prove your assertion. Justify your counterexample.

Expert Solution

Step 1

INTRODUCTION

Similar to basic operations on integers, set operations are a notion. Mathematical sets deal with a finite number of objects, such as letters, numbers, or any other real-world objects. There are instances when it becomes necessary to determine the connection between two or more sets. The idea of set operations is introduced here.

Given details: The proposed proof of the statement is:

$\forall A, B, C \subseteq U, C - (A \cup B) = (A^{c} - C) \cup (B^{c} - C)$

And the given proof is:

	Statement	Justification
1	$\forall A, B, C \subseteq U, C - (A \cup B) = (A^{c} - C) \cup (B^{c} - C)$	To be proved.
2	$C - (A \cup B) = (A \cup B) - C$	Commutativity
3	$= (A \cup B) \cap C^{c}$	Set difference law.
4	$= (A \cap C^{c}) U (B \cap C^{c})$	Distribution law.
5	$= (A - C) \cup (B - C)$	Set difference law.
6	$\forall A, B, C \subseteq U, C - (A \cup B) = (A^{c} - C) \cup (B^{c} - C)$	Transitivity of equality.

To determine:

(a) A counter example with three non-empty sets that shows that the statement to be proved is false. And justify the counter example.

(b) At least on statement in the proof that is not correct and create a counter example with non empty sets to prove the assertion. And justify the counter example.