Common Sets

Subset

$$A\subseteq B\leftrightarrow \forall x(x\in A\wedge x\in B)$$

Proper Subset

$$A⊊B\leftrightarrow (A\subseteq B)\wedge (A\ne B)$$

Superset

$$B\supseteq A$$

• $A$ is a Subset of $B$

Empty Set

• A Set that contains $0$ element
• Represented with $\varnothing$ or $\{\}$
• $\varnothing \notin \{\}$, but $\varnothing \subseteq \{\}$ and $\varnothing \in \{\varnothing \}$, where $\{\varnothing \}$ contains 1 element

Not a Null Set

Theorem 6.2.4

• An Empty Set is a s Subset of every Set

  
  

• Assume $A$ is all the possible sets, $\varnothing \subseteq A$

• Can be proved using Vacuous Truth of Universal

Singleton

• A Set with exactly one element

Mutually Disjoin Subset

• Also known as Pairwise Disjoint or Non-overlapping*
• Refer to Partition, the elements inside $\{A_1,A_2,A_3,A_4\}$ partition are Mutually Disjoin
• $A$ is called Union of Mutually Disjoint Subsets
• The collection of sets $\{A_1,A_2,A_3,A_4\}$ is said to be a partition of $A$

Disjoin Set

$$A\cap B=\varnothing$$

• Given 2 Set, both don’t have any elements in common

Partition

• A partition of Set $A$ is a finite or infinite collection of not empty, Mutually Disjoin Subset 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 $A$

Theorem 8.3.1

• Relation induced by Partition

• Two elements are related if and only if they belong to the Mutually Disjoin Subset in the partition. This connection created by the partition is called the relation induced by the partition

  
  

• 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

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

Power Set

• The power set of Set $A$ is all the possible Subset of $A$
• Given $A=\{x,y\}$, $\mathcal{P}(A)=\{\varnothing ,\{x\},\{y\},\{x,y\}\}$

Theorem 6.3.1

• The cardinality of Superset of finite set is $2^{cardinalityofthefiniteset}$