6. The following statements about sets are false. For eachstatement, give an example, i.e., a choice of sets, forwhich the statement is false. Such examples are calledcounterexamples. They are examples that are counter to,i.e., contrary to, the assertion.(a) AUBCAn B for all A, B.(b) ANØ:= A for all A.(c) AN (BUC) = (A ^ B) UC for all A, B, C.

Answered: 6. The following statements about sets…

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

6. The following statements about sets are false. For each
statement, give an example, i.e., a choice of sets, for
which the statement is false. Such examples are called
counterexamples. They are examples that are counter to,
i.e., contrary to, the assertion.
(a) AUBCAn B for all A, B.
(b) ANØ:
= A for all A.
(c) AN (BUC) = (A ^ B) UC for all A, B, C.

Expert Solution

Step 1

What is Counter Example:

A counterexample is an illustration that meets the criteria for a mathematical assertion but does not lead to the conclusion of the statement. Counterexamples are used to show a statement to be false. Identify the premise and conclusion of the given statement. The counterexample must be true in order for the hypothesis to be correct but the conclusion to be incorrect.

Given:

Given statements are:

For all $A, B$ , $A \cup B \subseteq A \cap B$ .

For all $A$ , $A \cap \emptyset = A$ .

For all $A, B, C$ , $A \cap (B \cup C) = (A \cap B) \cup C$ .