Class 8 – Set Theory

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Set Theory
  • Introduction to sets and subsets
  • Operations on sets
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Set Theory
  • Cardinal property of sets
  • Venn diagrams
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Introduction: The history of set theory is rather different from the history of most other areas of mathematics.  It is the creation of one person, George Cantor.

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

The objects in a set are called its members or elements.  By the term ‘ well – defined ’ we mean that it is defined in such a way that we are able to decide as to which object of the universe is there in our collection and which object is not there is our collection.

http://images.clipartpanda.com/bloc-clipart-note-md.pngNotation of a Set: We usually denote sets by capital letters and their elements by small letters.  If x is an element of a set A, we write, , which means ‘ x belongs to A’ or that ‘ x is an element of A’.  If x is not an element of set A, we say that ‘x does not belong to A’, and we write . It is customary to put the elements of a set within braces {   }.

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

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

Example:

i)     Let A be the set of vowels in English alphabet. Then A = { a, e, i, o, u }

ii)   Let B be the set of natural numbers between 3 and 10. Then,
B = {4, 5, 6, 7, 8, 9}

2. Rule Method (or 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} 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’

Example:

i) Let A= {41, 43, 47, 51, 53, 59}
Clearly A is the set of all prime numbers between 40 and 60.

Thus in the set-builder form, we write:

 A= {x/x is a prime number, 40 < x < 60}

i)     Let B= {8, 16, 24, 32, 40}. Then, in set-builder form, we write

     C = {x/is a multiple of 8, x < 40} or

     C = {x:x= 8n, and n < 6}

Types of Sets:

i)    Finite set: A set in which the process of counting of 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 if the process of counting of its elements does not come to an end.
Example: The set N = {1, 2, 3, 4, 5,6…….} of all natural numbers.

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 .
In Roster method, we denote it by {   }
Example:

http://images.clipartpanda.com/bloc-clipart-note-md.pngThe 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/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 = { x is an even between 13 and 19 number, x < 9} and B = { x is even,13 < x < 19}

Then A = {14, 16, 18} and B ={14, 16, 18}

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

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 = {21, 22, 23, 24, 25}. Then n(A)= 5

http://images.clipartpanda.com/bloc-clipart-note-md.pngCardinal number of empty set is zero. i.e., n(Ø) = 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, A ⇔ B

Thus, whenever n(A) =n(B), then we have  A ⇔ B

Example: Let A be the set of vowels in the English alphabet and B be the set of first five odd natural numbers. Then, A = {a, e,i, o, u} and B = {1, 3, 5, 7, 9}

Clearly n(A) = n(B)= 5 therefore A ⇔ B

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

Sub sets: The collections are generally linked in a given context. If we think of ourselves, then we belong to a certain society, which in turn belongs to a state, which in turn belongs to a country and so on. In the context of a school, all students of a school belong to a school. Some of them belong to a certain class. If there are sections with in a class, then some of these belongs to a section.

The need to have a mathematical relationship between different collections of similar types lead to the evolution of “subset”.

 

Subset: A set “A” is a subset of set “ B ”, if each member of “ A ” is also a member of set “ B ”.  We use symbol “ ⊂ ” to denote this relationship between a “subset” and a “set”. Hence,.

We read this symbolic representation as : set “A” is a subset of set “B”.  We express the intent of relationship as:

It is evident that set “B” is larger of the two sets. This is sometimes emphasized by calling set “B” as the “superset” of “A”. We use the symbol “ ⊃ ” to denote this relation:

If set “A” is not a subset of “B”, then we write this symbolically as:

 

Important results/deductions: Some of the important characteristic and related deductions are presented here:

Equality of two sets: In case, all elements of “B” are also in “A”, then two sets are equal. We express this symbolically as:

If  and , then A = B. This is true in other direction as well: If A = B, then  and .

We can write two instances in a single representation as:  and

The symbol “ ⇔ ”means that relation holds in either direction.

Relation with itself: Every set is a subset of itself. This is so because every element is present in itself.

Relation with Empty set: Empty set is a subset of every set. This deduction is a direct consequence of the fact that empty set has no elements. As such, this set is a subset of all sets.

Proper subset: We have seen from the deductions above that special circumstances of “equality” can blur the distinction between “set” and “subset”. In order to emphasize, mother-child relation between sets, we coin the term “proper subset”. If every element of set “ B ” is not present in set “ A” , then “A” is a “proper” subset of set“B”; otherwise not. This means that set “B” is a larger set, which besides other elements, also includes all elements of set “A”.

Set of vowels in English alphabet, “V”, is a “proper” subset of the set of English alphabet,

“E”. All elements of “V” are present in “E”, but not all elements of “E” are not present in “V”. Therefore, V⊂ E

Conventional differences. Some write a “proper” subset relation using symbol “ ⊂ ” and write symbol “ ⊆ ” to mean possibility of equality as well.

 

Number system: The number system is one such system, in which different number groups are related. Natural numbers are a subset of integers. Integers are a subset of rational numbers and rational numbers are a subset of real numbers. None of these sets are equal. Hence, relations are described by proper subsets.

 

The chain of relation among number sets is as follows:  (where C is the set of complex numbers).However, irrational numbers are also subsets of real numbers, but irrational number is not the same as rational numbers. We represent this relation by emphasizing that rational numbers is not a subset of irrational
Q (rational number) ⊄ Q1 (Irrational numbers). But irrational number is subset of real numbers. The real numbers comprises of only two subsets at the highest – rational and irrational.  Therefore, irrational numbers is the remaining collection after deducting rational numbers from real numbers.

Following the logic, we define set of irrational numbers as:

Q1 (irrational numbers) =

Power set: Power set is formed from of all possible subsets of a given set. It is denoted asP(A) (or), the collection of all subsets of a set “ A ” is called power set, P(A).

For example, consider a set given by:

                                                  A = {1, 3, 4}

What are the possible subsets? There are three subsets consisting of individual elements. { 1 }, { 3 } and { 4 }. Then, elements taken two at a time the subsets are: {1, 3}, {1, 4} and {3, 4}. Since order or sequence does not matter in set representation, there are only three subsets of two elements taken together. Now, the elements taken three at a time form only one subset: {1, 3,4}. Remember, a set is a subset of itself. Further, empty set (f) is a subset of any set. Hence, is also a subset of the given set “A”.

The set comprising of all possible subsets of a given set “A” is:

We note two important points from this representation of power set:

i)     The elements of a power set are themselves a set. In other words, every element of a power set is a set.

ii)   If the number of elements (cardinality) excludes empty set. It is, however, counted as members of power set. For a set having three elements, the total numbers of elements in the power set is:

iii) A set containing n elements has 2nsubsets but  proper subsets.

Example: The finite set “A” and “B” have “m” and “n” number of elements respectively. The total numbers of subsets of “A” is 56 more than the total number of subsets of“B”. Find “m” and “n”.

Solution: According to the relation obtained for power set, the total number of subsets of “A” and“B” are”

            

We need to find two equations to find “m” and “n”. For this we seek expansion of“56” in terms of powers of “2”.

56 = 8 × 7 = 8(8 – 1) = 23(23-1)

In order to get this form, we rearrange the expression on the LHS of the earlier equation as:

Equating powers of similar base,

n = 3and m = 6

Verification

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1.   Types of subsets:
i) Proper subsets.     ii) Empty sets.

2.   Emptyset is a subset of every set.

3.   The set itself is a subset of given set.