Class 7 – Set-Theory

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Set Theory
  • Review of set theory
  • Subsets
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Set Theory
  • Operations on sets
  • Venn diagrams
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Introduction: Set theory is the creation of George Cantor. Set theory is involved in mathematics and has important applications in other fields, like computer technology and atomic and nuclear physics.

Set: A well-defined collection of objects is called a set.

The elements of the set are the objects or members in a set. The term ‘well – defined’ means if it is possible to tell beyond doubt, about every object of the universe, whether it is there in our collection or not.

Notation of a Set: Usually we denote sets by upper-case letters and their elements by lower case letters.  The following notation is used to show a set of membership

 means that x is a member of A.

 means that x is not a member of A.

It is customary to represent the elements of a set within braces {   }.

Example: (i) A = {3, 17, 2} then

(ii) If A = {x/x is a prime number} then

Representation of a Set: There are two methods of representing a set.

I) Roster Method (or Tabulation Method): Under this method, we just make a list of the objects in our collection and put them within braces {   }.

Example:

(i) {1, 3, 5, 7…, 9007} is the set of odd counting numbers less than or equal to 9007.

(ii) {1, 2, 3 …} is the set of all counting numbers.

II) Rule Method (or Set-builder notation): Under this method, we list the property or properties satisfied by the elements of a set. We write, {variable/descriptive statement} or {x : x satisfies the properties of P} which means ‘The set of all those elements x such that each x satisfies the properties of P.

Each of the symbol ‘/’ and ‘:’ stands for ‘such that ’.

Set builder notation is frequently used when the roster method is either inappropriate or inadequate.

Example:

i)     Let A = {31, 37, 41, 43, 47}
Clearly A is the set of all prime numbers between 30 and 50.
Thus in the set-builder form, we write:
A = {x/x is a prime number, 30 < x < 50}

ii)   Let B = {7, 14, 21, 28, 35}. Then, in set-builder form, we write
B = {x/x is a multiple of 7, x < 40} or

Types of Sets:

i)     Finite set: A set in which the process of counting elements surely comes to an end is called a finite set.
Example: The set of all vowels in the English alphabet is a finite set having 5 elements namely a, e, i, o, u.

ii)   Infinite Set: A set which is not finite is called an infinite set. In other words, a set is called an infinite set the process of counting its elements does not come to an end.
Example: The set N= {1, 2, 3, 4, 5, 6…….} of all natural numbers is an infinite set.

iii) Empty set or Null set: A set having no elements at all is called an empty set or null set or void set and we denote it by f. In Roster method, we denote it by {   }.
Example:
(1) The set of even prime numbers greater than 2 is an empty set.
(2) The set is an empty set.

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The empty set is a finite set, since the number of elements contained in an empty set is 0.
Singleton Set: A set containing only one element is called a singleton set.

 

Example: {x/x is an even prime number} is a singleton set containing only one element namely 2.

Equal sets: Two sets A and B are said to be equal, if every element of A is in B and every element of B is in A and we write A = B.

 

Example: Let A = { is a composite number, x < 9}

and B = { is even, 3 < x < 9}

Then A = {4, 6, 8} and B = {4, 6, 8}

Clearly every element of A is in B and every element of B is in A.

                      A = B

Cardinal number of a set: The number of distinct elements contained in a finite set A is called its cardinal number and is denoted by n(A)

Example: Let A= {1, 2, 3, 6}. Then n(A) = 4

Since φ contains no element at all n(φ) = 0

Equivalent sets: Two finite sets A and B are said to be equivalent, if they have the same number of elements and we write,

Thus, whenever , then we have

 

Example: Let A be the set of vowels in the English alphabet and B be the set of first five natural numbers. Then, A= {a, e, i, o, u} and B = {1, 2, 3, 4, 5}

Clearly therefore

http://images.clipartpanda.com/bloc-clipart-note-md.pngTwo equal sets are always equivalent but equivalent sets need not to be equal.