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Chapter 2: Problem 54

State the law of set operations.

Short Answer

Expert verified
The law of set operations involves Union, Intersection, Difference, and Complement of sets. The Commutative Laws are: A ∪ B = B ∪ A and A ∩ B = B ∩ A. The Associative Laws are: (A ∪ B) ∪ C = A ∪ (B ∪ C) and (A ∩ B) ∩ C = A ∩ (B ∩ C). The Distributive Laws are: A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) and A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C). The Complement Laws include De Morgan's Laws: (A ∪ B)' = A' ∩ B' and (A ∩ B)' = A' ∪ B', Double Complement Law: (A')' = A, and complements of Universal and Empty Sets: U' = ∅ and ∅' = U.

Step by step solution

01

1. Define Set Operations

A set operation is a mathematical function that takes two sets as input and produces a resultant set derived from the input sets. Let A, B, and C be three sets, the fundamental set operations include: a. Union: The union of sets A and B, denoted by A ∪ B, is the set of all elements that belong to A or B or both. b. Intersection: The intersection of sets A and B, denoted by A ∩ B, is the set of all elements that belong to both A and B. c. Difference: The difference of sets A and B, denoted by A - B, is the set of all elements that belong to A but not to B. d. Complement: The complement of set A, denoted by A', is the set of all elements that do not belong to A.
02

2. Commutative Laws

The commutative laws of set operations are: a. Union: A ∪ B = B ∪ A b. Intersection: A ∩ B = B ∩ A
03

3. Associative Laws

The associative laws of set operations are: a. Union: (A ∪ B) ∪ C = A ∪ (B ∪ C) b. Intersection: (A ∩ B) ∩ C = A ∩ (B ∩ C)
04

4. Distributive Laws

The distributive laws of set operations are: a. Union over Intersection: A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) b. Intersection over Union: A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
05

5. Complement Laws

The complement laws of set operations are: a. De Morgan's Laws: i. (A ∪ B)' = A' ∩ B' ii. (A ∩ B)' = A' ∪ B' b. Double Complement Law: (A')' = A c. Complement of Universal Set: U' = ∅ (empty set) d. Complement of Empty Set: ∅' = U (universal set)

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