Definition
Set
A collection of objects with specific characteristics is called a set.
\(A=\{1,2,3,4,5\}\)
\(A=\{x\mid0<x<6,\ x\in\mathbb{Z}\}\)
Empty Set
A set which does not have any elements is called an empty set. It is denoted with a \(\varnothing\) or \(\{\ \}\) symbol.
Equal Set
Sets which consist of the same elements are called equal sets.
\(\left.\begin{aligned}A=\{1,2,3\}\\B=\{x\mid1\le x\le3,\ x\in\mathbb{Z}\}\end{aligned}\right\}\)
\(\Rightarrow A=B\)
- \(n(A)\Rightarrow\text{is the number of elements of set A}\)
- \(a\in A\Rightarrow\text{a is an element of set A}\)
- \(a\notin A\Rightarrow\text{a is not an element of set A}\)
Equivalent Set
\(n(A)=n(B)\Rightarrow A\cong B\)
\(A=\{1,2,3\}\)
\(B=\{a,b,c\}\)
Universal Set
The set, on which operations are performed and which contains all sets is called the universal-set and is denoted with U.
Subset
Let A and B be two sets. Of every element of set B is also an element of set A then it said that set B is a subset of set A, or that set A spans set B. This is notated as \(A\supset B\) or \(B\subset A\).
\(\left.\begin{aligned}A=\{a,b,c,d,e\}\\B=\{b,c,e\}\end{aligned}\right\}\)
\(\Rightarrow B\subset A\text{ or }A\supset B\)
Let A, B and C are any sets and U is the universal set
- \(\varnothing\subset A\)
- \(A\subset A\)
- \(A\subset U\)
- \(A\subset B\text{ and }B\subset A\Rightarrow A=B\)
- \(A\subset B\text{ and }B\subset C\Rightarrow A\subset C\)