Class 6 – Set-Theory

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

A collection of objects is said to be well defined if it is possible to tell beyond doubt as to which object of the universe is in our collection and which is not there in the collection.

In our mathematical language, everything in this universe, whether living or non-living is called an object.

 

Explanation:

1.   The objects in a set are called its members or elements.

2.   We usually represent these members or elements by small letters within braces
{  } and the set representing these members or elements by capital letters.

3.   Here x ∈ A means that element x belongs to set A and  x ∉ A means that element x does not belong to set A.

4.   If adjectives have been used to describe the objects then they do not form a set and they do not form a well-defined collection.

 

Example: “Good players”, “honest persons”, “intelligent boys”, “beautiful girls”, “tall boys”, etc do not form sets.

Example:

1. Collection of first six odd numbers is the set containing the elements. 1, 3, 5, 7, 9,

Let us call this set as A. Then,
A = {1, 3, 5, 7, 9, 11}
Clearly 1 ∈ A, 3 ∈ A, 5 ∈ A, 7 ∈ A. 9 ∈ A, 11 ∈ A.
but 2 ∉ A, 4 ∉ A, 6 ∉ A, etc.

2. Consider the collection of beautiful buildings. There is no specified definition to be beautiful. Hence, the collection of beautiful buildings is not a set.

 

Representation of a set:

There are two methods of representing a set.

1.   Listing method (Roster form)

2.   Rule-method (Set – builder form)

1.   Listing method (Roster form):

Under this method, we just make a list of the members of the set and put them within braces.

Example: Let A be the set of all digits less than 7then A = {0, 1, 2, 3, 4, 5, 6}

 

2.   Rule-Method (Set-builder form):

Under this method, we list the property or properties satisfied by the elements of a set.

We write {x/x satisfies the properties p}, which means “the set of all those x such that each x satisfies the properties p.

The symbol ‘/’ stands for ‘suchthat’. Sometimes we use the symbol ‘:’ for ‘such that’.

 

Example: A= {x/x< 4 and x ∈ N} is the set builder form of the set
A = {1,2,3}

Cardinal numberof a set: The number of distinct elements in a set A is called the cardinal number of A, to be denoted by n(A).

Example: LetA = {5, 7, 13, 17, 19}. Then n(A) = 5

 

Finite andinfinite sets: A finite set contains a definite number ofelements. Sets which are not finite are called infinite sets.

Empty set orNull set or Void set: A set having no element at all is called an empty set or a null set or a void set.

It is denoted by φ(read as pie). In Roster form, we write φ = { }. Clearly n(φ)= 0.

Example: A = {x/x is a whole number, 4 < x< 5}. Clearly there is no whole number between 4 and 5. ∴ A = φ (or){ }

 

Non-empty set: A set containing at least one element is called a non-empty set.

Thus ‘A’ is non-empty, if A ≠ φ.

Singleton set: A set containing exactly one element is calleda singleton set. Thus, A is a singleton set, if n(A) = 1.

Example: A = {x/x is an even prime}

⇒ A = {2}

Therefore A is a singleton set.

 

Equivalentsets: Two sets A and B are said tobe equivalent if their cardinalities are same i.e. n(A) = n(B).

And we write A ↔ B.

Example: A ={Factors of 8}; B = {Factors of 6}

⇒ A = {1, 2, 4, 8}; B = {1, 2, 3, 6}

⇒ n(A) = 4; n(B) = 4

⇒ n(A) = n(B)

Hence A and B are equivalent sets or A ↔ B.

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

Example: {a, b, c} = {c, a, b}. The two sets are equal.

Example: A= {x/x = all prime numbers between 2and 6}

B = {x/x = all odd numbers between1 and 7}

A = {3, 5}

B = {3, 5}

⇒ A = B

⇒ n(A) = n(B)

⇒ A ↔ B

Hence equal sets are equivalent, butequivalent sets are may are may not be equal.

Operations on sets:

Union of sets: The union of two sets A and B, denoted by  (i.e., A Union B) is the set of all those elements which are either in A or B or in both A and B.

Example: Let A = {1, 2, 3, 4, 5} and B = {4, 5,6, 7} then  =?
Solution: = {1, 2,3, 4, 5, 6, 7}

 

Example: Let A = {Ram, Prem, Sonu} and B = {Monu,Sonu, Sheela}, then find .
Solution: ={Ram, Prem, Sonu, Monu, Sheela}.

 

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1.  In order to form  from two given sets A and B,we first take all those elements and then take only those elements of B, which are not there in A

2.  The union of elements can be irrespective of order

3.  The symbol ‘∈’indicates belongs to and ‘∉’ symbol indicates doesn’t belong to.

 

Intersection of sets: The intersection of two setsA and B, denoted by  is the set of all those elements, which are common to both A and B.
Thus,  =

Example: LetA = {1, 3, 5, 7, 9} and B = {2, 3, 7, 10} then =?
Solution:A = {1, 3, 5, 7, 9}, B = {2, 3, 7, 10}
 = {3, 7}

 

Example: Let A= {multiples of 3 less than 24} and B = {multiples of 4 less than 28}
Solution: A = {multiples of 3 lessthan 24}
A = {3, 6, 9, 12, 15, 18, 21}
B = {multiples of 4 less than 28}
B= {4, 8, 12, 16, 20, 24}
 = {12}.

Definition of subset: If A and B are two sets suchthat every element of A is in B, then we say that A is a subset of B and we write, A ⊆ B.

If there exists at least one element in A which is not a member of B, then we say that A is not a subset of B and we write .

Superset: Whenever a set A is a subset of set B, we saythat B is a superset of A and we write, B ⊇ A.

Example:

Let A = {Set of letters of the English alphabets}

B = {Set of vowels of the English alphabet}

The A = {a, b, c , ………..z}

B = {a, e, i, o, u}

⇒ B ⊆A or A ⊇ B.

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i) Every set is a subset of itself.

ii) Empty set is a subset of every set.

 

Proper subset: If A and B are two sets such that A is asubset of B and A ≠ B, then we say that A is a proper subset of B and we write, A ⊂ B.

Example: Set A = {2, 4, 6, 8, 10}; B = {4, 6, 8}

Then every element of B is in A and n(B)< n(A) then B ⊂ A.

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i) No set is a proper subset of itself

ii) φ is aproper subset of every set except itself.

Number of subsets of a given set: A set A containing n elements has 2n subsets denoted by n[P(A)].

Example:

i)     A setcon taining no element has 20 = 1 subset which is the empty set itself.

ii)   A setcon taining a single element or a singleton set has 21 = 2 subsets;the empty set and the given set itself.

iii) A = {a, b, c}, listall subsets of A and verify through calculation.
Subsets of A = {{a}, {b}, {c}, {a,b}, {b,c}, {c, a}, {a,b,c}, φ}

Verification: Numberof subsets of A = 23 = 2 × 2 × 2 = 8

Number ofproper subsets: A set A containing n elements has (2n- 1) proper subsets represented by n[P(A)]-1.

Example: C = {p, q, r}

Therefore number of possible subsets of C = 2n – 1 = 23 – 1 = 7

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1.   Disjoint Sets: Two sets having no element in common are known as disjoint sets.
Two sets A and B are said to be disjoint sets, if , i.e., there is no element common to A and B.
Example: The set of all odd numbers and the set of all even numbers are disjoint sets.

2.   Overlapping Sets: Two setsA and B are called overlapping sets, if they have at least one element common in them.
Example: Let A be the set of first 5prime numbers and B be the set of first 5 multiples of  2. Then, A = {2, 3, 5,7, 11} and B = {2, 4, 6, 8, 10}.
Clearly, A and B have one element in common namely, 2.
So, A and B are overlapping sets.
If , then Aand B are called intersecting or overlapping sets, if they have at  least one element in common.
Example: Let A = {2, 3, 5, 7, 11}, B= {2, 4, 6, 8, 10} and C = {1, 3, 5, 7, 9} Then,  and
Thus,two sets A and B are intersecting sets and B and C are disjoint sets.