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Sets




1. Introduction

What is Sets?
Simply put, it's a collection of objects
Examples
N : the set of all natural numbers
Z : the set of all integers
Q : the set of all rational numbers
R : the set of real numbers
Z+ : the set of positive integers
Q+ : the set of positive rational numbers, and
R+ : the set of positive real numbers
Set of even numbers: {..., -4, -2, 0, 2, 4, ...}
Set of odd numbers: {..., -3, -1, 1, 3, ...}
Set of prime numbers: {2, 3, 5, 7, 11, 13, 17, ...}
Positive multiples of 3 that are less than 10: {3, 6, 9}

2. Methods of representing a set

For sets, we simply put each element, separated by a comma, and then put some curly brackets around the whule thing.
Sets are usually denoted by capital letters A, B, C, X, Y, Z, etc
The elements of a set are represented by small letters a, b, c, x, y, z, etc
When we say an element a is in a set A, we use the symbul \( \in \) to show it.
And if something is not in a set use \( \notin \)
Example: In a set of even number \(E\), \(2 \in E\) but \(3 \notin E\)
Two Methods are used to represent Sets

(a) Roster forms

In a Roster forms, all the elements in the set is listed.
Example
Set of Vowel ={ a,e,i,o,u}
Some Important points
  • In roster form, the order in which the elements are listed is immaterial
  • while writing the set in roster form an element is not generally repeated

(b) Set Builder Form

  • In set-builder form, all the elements of a set possess a single common property which is not possessed by any element outside the set.For example, in the set \(\left\{ {2,4,6,8} \right\}\), all the elements possess a common property, namely, each of them is a even number less than 10. Denoting this set by \(N\), we write
    \(N = \left\{ {x:x{\rm{ is a even number less 10}}} \right\}\)
  • We describe the element of the set by using a symbul x (any other symbul like the letters y, z, etc. could be used) which is fullowed by a culon ":" . After the sign of culon, we write the characteristic property possessed by the elements of the set and then enclose the whule description within braces


3. Types of sets

(a) Empty set

  • A set which does not contain any element is called the empty set or the null set or the void set
  • It is denoted by \(\phi \) or \(\left\{ {} \right\}\)
  • It is a set with no elements
  • Examples of empty sets is
    \(D = \left\{ {x:{x^2} = 9,x{\rm{ is even}}} \right\}\)
    Here D is the empty set, because the equation \({{x^2} = 9}\) is not satisfied by any even value of x

(b) Finite or infinite set

  • If \(M\) is a set then \(n\left( M \right)\) defines the number of distinct elements in the set M.
  • If \(n\left( M \right)\) is zero or finite ,then \(M\) is a finite set
  • If \(n\left( M \right)\) is infinite then \(M\) is a infinite set

(c) Equal sets

  • Two sets are said to be equal if they have same members in them.
  • For \(A\) and \(B\) to be equal, every member of \(A\) should be present in set \(B\) and every member of \(B\) to be present in set \(A\)
  • It is denoted by equality sign \(A = B\)


4. Subset and Proper Subset

  • A set \(A\) is said to be a subset of a set \(B\) if every element of \(A\) is also an element of \(B\) .
  • It is denoted by
    \(A \subset B\) if whenever \(a \in A\) , then \(a \in B\)
  • If \(A \subset B\) and \(B \subset A\) , then \(A = B\).
  • Every set is subset of itself \(A \subset A\)
  • Empty set is subset of every set \(\phi \subset A\)
  • If \(A \subset B\) and \(A \ne B\) , then \(A\) is proper subset of \(B\). In such a case \(B\) is called superset of set \(A\)

5. Subset of set of the real numbers

N : the set of all natural numbers
Z : the set of all integers
Q : the set of all rational numbers
R : the set of real numbers
Z+ : the set of positive integers
Q+ : the set of positive rational numbers, and
R+ : the set of positive real numbers
\(T = \left\{ {x:x \in R{\rm{ and }}x \notin Q} \right\}\), i.e., all real numbers that are not rational
\(N \subset Z \subset Q,{\rm{ }}Q \subset R,{\rm{ }}T \subset R,{\rm{ }}N \not\subset T\)

6. Interval as subset of R Real Number

\(a,b \notin R,b > a\)
\((a,b)\) It is the open interval set between point and b such that All the points between a and b belong to the open interval (a, b) but a, b themselves do not belong to this interval \(\{ y:a < y < b\} \)
\([a,b]\) It is the closed interval set between point and b such that All the points between a and b belong to the open interval (a, b) including a, b \(\{ x:a \le x \le b\} \)
\([a,b)\) It is the open interval set between point and b such that All the points between a and b belong to the open interval (a, b) including a, but not b \(\{ x:a \le x < b\} \)
\((a,b)\) It is the open interval set between point and b such that All the points between a and b belong to the open interval (a, b) including b, but not a \(\{ x:a < x \le b\} \)



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