Set Partition

Abstract

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• A partition of Set $A$ is a finite or infinite collection of NOT Empty Set, Mutually Disjoin Set that can be chained with OR to form $A$
  

$$\forall x\in A,\exists !S\in \zeta (x\in S)$$

• $S$ is one of the mutually disjoin subset, also known as component of the partition
• $\zeta$ is the partition
• So basically each $S$ isn’t empty, and its elements are not in other mutually disjoin subset

Subset can contain duplicate elements

Given a set $A=\{a,b,c,d,e,f\}$ ,

$\{\{a\},\{b,f,b\},\{c,e,d,e\}\}$ is a valid partition, because it covers all the elements inside set $A$

Relation from Partition

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$$\forall x,y\in A,xRy\leftrightarrow \exists \text{ a component of }S\text{ of }\zeta \text{ s.t. }x,y\in S$$

• Theorem 8.3.1, Relation induced by Set Partition
• Two elements are related if and only if they belong to the Mutually Disjoin Set in the partition. This connection created by the partition is called the relation induced by the partition

Important

• Let $A$ be a Set with a Partition
• Let $R$ be the relation induced by the partition
• Then $R$ is Reflexive, Symmetric and Transitive

Example

Example

Imagine dividing students in a class into groups based on their favourite sport. The relation induced by this partition would tell us which students share the same sports preference.