Sets - Chapter Summary

Sets

In our daily life, we have to deal with the collection of objects of one kind or the other.
Eg: 

• The collection of even natural numbers less than 12. (2,4,6,8,10)
• The collection of vowels in English alphabets. (a,e,i,o,u)
• The collection of well known mathematicians in the world might not be the same for everyone. So it is not a set.

Therefore we can just define set as - A set is a well-defined collection of objects.

Sets can be represented by two ways: 

• Roster or Tabular Form - All the elements of a set are listed separated by commas and are enclosed within braces { }.Elements are not repeated generally.  Examples: A={1,2,3,4,5} or B={5,10,15,....}     

• Set builder Form - In set-builder form, set is denoted by stating the properties that its members satisfy. Examples: A={x:x is a natural number less than 6} or B={x:x is a multiple of 5}

Empty Set

Empty set is the set having no elements in it. It is denoted by { } or ∅. Empty Set is also called null set or void.

On the basis of number of elements sets are of two types: 

• Finite Sets - Finite set is a set in which there are definite number of elements.  or { } or Null set is a finite set as it has 0 number of elements which is a definite number. Examples: A={1,2,3,4,5} or B={x:x is an alphabet in CBSEONLINESTUDY}
• Infinite Sets - A set that is not finite is called infinite set. Examples: B={x:x is an even number} or B={1,2,3,4.........}

Nativity of an Element to a Set

For representing 2 as a member of set A, we can use the symbol '∈'. '2∈A' and we read it as '2 belongs to the set A'. To denote -1 is not a member of set A, we represent it as '-1∉A' and we read it as minus 1 not belongs to A

Equal Sets

Two sets A and B are equal if they have exactly the same elements. If they are equal, we write it as A=B and if they are not we write them as A≠B. A set does not change if one or more elements of the set are repeated.
A and B are equal even if:
A={1,2}
B={11,22}

Subset

Set 'A' is said to be the subset of Set 'B' if every element of Set 'A' is also there in Set 'B'. It is written as 'A⊂B' and we read it as "A subset of B".

• If A⊂B and B⊂A, then the sets A and B are equal.
• Every set is a subset of itself.
• Empty set is a subset of every set.

Union of Sets

The union of sets 'A' and 'B' consists of elements which are either in A or in B. We can denote the union of sets 'A' and 'B' as 'AUB' and read it as "A union B".

AUB={x:x belongs to A or belongs to B}

Example:

A={1,2,3,4}

B={4,5,6,7}

AUB={1,2,3,4,5,6,7} 

Properties of Union of Sets

• AUB = BUA - Commutative Property
• (AUB)UC = AUBUC - Associative Property
• AU∅ = A - Law of Identity
• AUA = A - Idempotant law
• U(Universal Set)UA = U 

Intersection of Sets

The Intersection of two sets A and B is the set of all elements which are common to the sets A and B. The symbol ∩ is used to denote intersection. Symbolically we write A∩B={x:x is an element of A and x is an element of B}

Properties of Intersection of Sets

• A∩B = B∩A - Commutative Property
• (A∩B)∩C = A∩∩C - Associative Property
• ∅∩A = ∅
• U∩A = A
• A∩A = A - Idempotent Law
• (A∩B)UC = (A∩B)U(A∩C)U(B∩C)

Two sets are said to be disjoint if A∩B=∅

Difference of Two Sets

The difference of set A and B is the set of elements which belongs to A ut not belong to B. Symbolically we write:
A-B={x:x belongs to A but don't belong to }

Complement of a Set

Let U e universal set and A, its subset. The complement of A is the set of all elements which belongs to U but are not the elements of A. Symbolically we write AI and read it as  A complement.  AI={x:x belongs to U but don't belong to A}

Properties of Compliment of Sets

• AUAI = U
• A∩AI = ∅
• (AUB)I = AI∩BI
• (A∩B)I = AI U AI
• (AI)I = A
• UI = ∅
• ∅I = U

n(AUB) = n(A) + n(B) - n(A∩B)

if n(A∩B) = ∅

n(AUB) = n(A) + n(B)