Which of the following are true for all sets A, B, and C. If the statement is true, prove it by showing each is a subset of the other. If not, provide a counterexample.(d) (A')' = A(e) A \ B = (B \ A)'(f) (A \ B) ∩ (B \ A) = ∅(g) If A ∩ B = ∅, then A ⊆ B. ### Set Theory PropertiesThe following are properties related to basic set theory operations:(d) \((A')' = A\) - This states that the complement of the complement of set \(A\) is \(A\) itself.(e) \(A \setminus B = (B \setminus A)'\) - This equation means the difference between sets \(A\) and \(B\) is equal to the complement of the difference between sets \(B\) and \(A\).(f) \((A \setminus B) \cap (B \setminus A) = \emptyset\) - This denotes that the intersection of the difference between sets \(A\) and \(B\), and the difference between sets \(B\) and \(A\), is the empty set.(g) If \(A \cap B = \emptyset\), then \(A \subseteq B\). - This indicates that if the intersection of sets \(A\) and \(B\) is empty, then \(A\) is a subset of \(B\).### Explanation:Each property describes relations and operations between sets using symbols:- \(A'\): Complement of set \(A\).- \(\setminus\): Set difference.- \(\cap\): Intersection of sets.- \(\emptyset\): The empty set.- \(\subseteq\): Subset relation, indicating every element of \(A\) is also in \(B\).

(d) Let, x ∈ (A')' => x ∉ A' => x ∈ A So, (A')' ⊆ A. . . . . . (I)…

Which of the following are true for all sets A, B, and C. If the statement is true, prove it by showing each is a subset of the other. If not, provide a counterexample. (d) (A')' = A (e) A \ B = (B \ A)' (f) (A \ B) ∩ (B \ A) = ∅ (g) If A ∩ B = ∅, then A ⊆ B.

Which of the following are true for all sets A, B, and C. If the statement is true, prove it by showing each is a subset of the other. If not, provide a counterexample. (d) (A')' = A (e) A \ B = (B \ A)' (f) (A \ B) ∩ (B \ A) = ∅ (g) If A ∩ B = ∅, then A ⊆ B.

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

Which of the following are true for all sets A, B, and C. If the statement is true, prove it by showing each is a subset of the other. If not, provide a counterexample.

(d) (A')' = A
(e) A \ B = (B \ A)'
(f) (A \ B) ∩ (B \ A) = ∅
(g) If A ∩ B = ∅, then A ⊆ B.

### Set Theory Properties

The following are properties related to basic set theory operations:

(d) \((A')' = A\)
- This states that the complement of the complement of set \(A\) is \(A\) itself.

(e) \(A \setminus B = (B \setminus A)'\)
- This equation means the difference between sets \(A\) and \(B\) is equal to the complement of the difference between sets \(B\) and \(A\).

(f) \((A \setminus B) \cap (B \setminus A) = \emptyset\)
- This denotes that the intersection of the difference between sets \(A\) and \(B\), and the difference between sets \(B\) and \(A\), is the empty set.

(g) If \(A \cap B = \emptyset\), then \(A \subseteq B\).
- This indicates that if the intersection of sets \(A\) and \(B\) is empty, then \(A\) is a subset of \(B\).

### Explanation:

Each property describes relations and operations between sets using symbols:
- \(A'\): Complement of set \(A\).
- \(\setminus\): Set difference.
- \(\cap\): Intersection of sets.
- \(\emptyset\): The empty set.
- \(\subseteq\): Subset relation, indicating every element of \(A\) is also in \(B\).

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