Sets

Sets

Definition

A set is a collection of distinct elements. For example, the set of natural numbers less than 5 can be written as:

$$A=\{0,1,2,3,4\}$$

Universal Set

A universal set $U$ is a set that contains all the objects under consideration. For example, if we are discussing the set of all integers, then:

$$U=\mathbb{Z}$$

where $\mathbb{Z}$ represents the set of all integers.

Subset

A subset is a set where all its elements are also contained within another set. For example, if:

$$A=\{1,2,3\}$$

then:

$$B=\{1,2\}$$

is a subset of $A$, which is written as:

$$B\subseteq A$$

Unions of Sets

The union of two sets $A$ and $B$ is a set containing all elements that are in either $A$, $B$, or both. For example:

$$A=\{1,2,3\}$$ $$B=\{3,4,5\}$$

The union of $A$ and $B$ is:

$$A\cup B=\{1,2,3,4,5\}$$

Intersection

The intersection of two sets $A$ and $B$ is a set containing all elements that are common to both $A$ and $B$. For example:

$$A=\{1,2,3\}$$ $$B=\{2,3,4\}$$

The intersection of $A$ and $B$ is:

$$A\cap B=\{2,3\}$$

Set Difference

Let $A$ and $B$ Be two sets. They are non empty. Then the Symmetric difference is defined as $A-B=\{x\mid x\in A\text{ and }x\notin B\}$ Similarly, the set difference between $B$ and $A$ is: $B-A=\{x\mid x\in B\text{ and }x\notin A\}$

Symmetric Difference

Let $A$ and $B$ be two non empty sets. Then the symmetric difference between $A$ and $B$ will be. $A\oplus B=\{x|x\in Aorx\in B\}$ Mathematically it can be represented with. $A\oplus B=(A-B)\cup (B-A)$ Or $A\oplus B=(A\cup B)-(A\cap B)$

Powersets

The power set of a set ( S ) is the set of all possible subsets of ( S ), including the empty set and ( S ) itself. It is denoted by ( \mathcal{P}(S) ) or ( 2^S ).

Definition

If ( S ) is a set, then the power set of ( S ) is defined as:

$$\mathcal{P}(S)=\{T\mid T\subseteq S\}$$

This means that ( \mathcal{P}(S) ) contains every subset ( T ) such that ( T ) is a subset of ( S ).

Example

Let ( S = {a, b} ). The power set of ( S ) is:

$$\mathcal{P}(S)=\{\varnothing ,\{a\},\{b\},\{a,b\}\}$$

Properties

1. Cardinality: The number of elements in the power set is ( 2^n ), where ( n ) is the number of elements in ( S ). For instance, if ( S ) has 2 elements, the power set will have ( 2^2 = 4 ) elements.

2. Inclusion: The empty set ( \emptyset ) and the set ( S ) itself are always included in the power set.

3. Subset Relationship: Every subset of ( S ) is an element of ( \mathcal{P}(S) ).

LaTeX Representation

In LaTeX, the power set is represented as:

\documentclass{article}
\usepackage{amsmath}
 
\begin{document}
 
The power set of a set \( S \) is defined as:
$$
\mathcal{P}(S) = \{ T \mid T \subseteq S \}
$$
 
For example, if \( S = \{a, b\} \), then:
$$
\mathcal{P}(S) = \{ \emptyset, \{a\}, \{b\}, \{a, b\} \}
$$
 
If a set \( S \) has \( n \) elements, then the power set \( \mathcal{P}(S) \) has \( 2^n \) elements.
 
\end{document}

Cardinality of a Set

The cardinality of a set refers to the number of elements in the set. For a finite set, this is simply the count of elements within the set. The cardinality of a set ( A ) is denoted by ( |A| ).

Example

For a set ( A ) defined as:

$$A=\{1,2,3\}$$

The cardinality of ( A ) is:

$$|A|=3$$

This indicates that the set ( A ) contains 3 elements.

Partition

Let $S$ be a non-empty set. A partition of $S$ refers to the division of $S$ into disjoint subsets such that the union of these subsets reconstructs the original set $S$.

In other words, a partition of $S$ is a collection of non-empty subsets $\{S_1,S_2,\dots ,S_n\}$ such that:

1. $S_i\cap S_j=\varnothing$ for all $i\ne j$ (the subsets are pairwise disjoint).
2. ${\bigcup }_{i=1}^n{S}_i=S$ (the union of all subsets equals the original set $S$).

Example

Consider the set $S=\{1,2,3,4\}$. One possible partition of $S$ is:

$$\{\{1,2\},\{3,4\}\}$$

Here, the subsets $\{1,2\}$ and $\{3,4\}$ are disjoint and their union is $S$:

$$\{1,2\}\cup \{3,4\}=\{1,2,3,4\}=S$$

Properties of Sets

1. Idempotent Laws

   • For any set $A$: $$A\cup A=A$$ $$A\cap A=A$$

2. Domination Laws

   • For any set $A$ and the universal set $U$: $$A\cup U=U$$ $$A\cap U=A$$ $$A\cup \varnothing =A$$ $$A\cap \varnothing =\varnothing$$

3. Complement Laws

   • For any set $A$ and its complement $A^c$: $$A\cup A^c=U$$ $$A\cap A^c=\varnothing$$

4. De Morgan’s Laws

   • For any sets $A$ and $B$: $$(A\cup B{)}^c=A^c\cap B^c$$ $$(A\cap B{)}^c=A^c\cup B^c$$

5. Distributive Laws

   • For any sets $A$, $B$, and $C$: $$A\cup (B\cap C)=(A\cup B)\cap (A\cup C)$$ $$A\cap (B\cup C)=(A\cap B)\cup (A\cap C)$$

6. Associative Laws

   • For any sets $A$, $B$, and $C$: $$(A\cup B)\cup C=A\cup (B\cup C)$$ $$(A\cap B)\cap C=A\cap (B\cap C)$$

7. Commutative Laws

   • For any sets $A$ and $B$: $$A\cup B=B\cup A$$ $$A\cap B=B\cap A$$

Cartesian Product

The Cartesian product of two sets is a fundamental concept in set theory and mathematics. It refers to the set of all possible ordered pairs that can be formed by taking one element from each of the sets.

Definition

If $A$ and $B$ are two sets, then the Cartesian product of $A$ and $B$, denoted by $A\times B$, is defined as:

$$A\times B=\{(a,b)\mid a\in A\text{ and }b\in B\}$$

Here, $(a,b)$ represents an ordered pair where $a$ is an element from set $A$ and $b$ is an element from set $B$.

Example

Consider the sets $A=\{1,2\}$ and $B=\{x,y\}$. The Cartesian product $A\times B$ is:

$$A\times B=\{(1,x),(1,y),(2,x),(2,y)\}$$

In this example, we form ordered pairs by pairing each element of $A$ with each element of $B$.

Properties

1. Order Matters: The Cartesian product $A\times B$ is not the same as $B\times A$. For example, $A\times B$ contains pairs where the first element is from $A$ and the second is from $B$, while $B\times A$ contains pairs where the first element is from $B$ and the second is from $A$.

2. Empty Set: If either $A$ or $B$ is an empty set, then their Cartesian product will also be empty. For instance, if $A=\varnothing$, then $A\times B=\varnothing$ for any set $B$.

3. Associativity: The Cartesian product is associative, meaning that for three sets $A$, $B$, and $C$, the following holds:

$$(A\times B)\times C=A\times (B\times C)$$

However, the interpretation of the pairs differs.

4. Number of Elements: If $A$ has $m$ elements and $B$ has $n$ elements, then $A\times B$ will have $m\times n$ elements.

LaTeX Representation

In LaTeX, the Cartesian product is represented as:

\documentclass{article}
\usepackage{amsmath}
 
\begin{document}
 
The Cartesian product of two sets \( A \) and \( B \) is defined as:
$$
A \times B = \{ (a, b) \mid a \in A \text{ and } b \in B \}
$$
 
For example, if \( A = \{1, 2\} \) and \( B = \{x, y\} \), then:
$$
A \times B = \{ (1, x), (1, y), (2, x), (2, y) \}
$$
 
\end{document}
 
## References