(a) Explanation Given information: Set: { 1 , 2 , 3 } Formula used: 1) Partition: A separation of a non-empty set A into mutually disjoint non-empty subsets is called a partition of set A . 2) Disjoint: Two sets A and B are called disjoint if and only if A ∩ B = ϕ . 3) Union: If A and B are sets, the union of A and B is the set A ∪ B ( read as‘ A union B ’) given by A ∪ B = { x | x ∈ A o r x ∈ B } . 4) Intersection: If A and B are sets, the intersection of A and B is the set A ∩ B ( read as‘ A intersection B ’) given by, A ∩ B = { x | x ∈ A a n d x ∈ B } . Calculation: Let A = { 1 , 2 , 3 } Partition 1: X 1 = { 1 } , X 2 = { 2 } , X 3 = { 3 } X 1 ≠ ϕ , X 2 ≠ ϕ By using definition of union of sets, X 1 ∪ X 2 ∪ X 3 = { 1 , 2 , 3 } = A By using definition of intersection of sets X 1 ∩ X 2 = ϕ , X 2 ∩ X 3 = ϕ , X 1 ∩ X 3 = ϕ . By definition of partition, Subset X 1 , X 2 and X 3 forms a partition of A (b) To determine To find: All different partitions of the set { 1 , 2 , 3 , 4 } .

8th Edition

ISBN: 9781285965918

Author: Gilbert

Publisher: Cengage

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Set Theory

Every field in the present world involves a set. The theory of sets was developed by German mathematician Georg Cantor while working on the “problems of trigonometric series”.

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Chapter 1.1, Problem 9E

(a)

To determine

To calculate: All different partitions of the set {1, 2, 3}.

(b)

To determine

To find: All different partitions of the set {1, 2, 3, 4}.

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Chapter 1.1, Problem 8E

Chapter 1.1, Problem 10E

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Elements Of Modern Algebra

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All different partitions of the set { 1 , 2 , 3 } .

Solution Summary: The author explains how to calculate the different partitions of the set left1,2, 3right.

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To calculate: All different partitions of the set {1, 2, 3}.

To find: All different partitions of the set {1, 2, 3, 4}.

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Chapter 1 Solutions