Handwritten notes on set theory for Bsc 1st semester.

Handwritten notes on set theory for BSc 1st semester.

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Free Handwritten Study Materials for BSc  Mathematics Students DDU.

 

First, let's discuss the concept of set theory which provides a more convenient way to understand the related examples and previous year's questions of set theory. No doubt, without concepts, any solutions to equations may be tough for you.

Let's understand one by one.

All these topics are based on the New syllabus of DDU BSC 1st semester and first paper.

What do you mean by sets?

Set means "collection", "aggregate", and "class". A well-defined collection of objects is known as a set.

Example:

1. The collection of vowels in English alphabets.
2. The collection of even numbers is less than 100.
3. The collection of all states in the Indian Union.
4. The collection of past prime ministers of the Indian Union.
5. The collection of maths books in a library.
   

To make it more convenient to frequently interact with some related examples of sets, we have reserved some letters for these sets as listed below:

Letters Symbols used in Number System

• N→ For the set of natural numbers.
• Z→ For the set of integers.
• Z⁺→ For the set of all positive integers.
• Q→ For the set of all rational numbers.
• Q⁺→ For the set of all positive rational numbers.
• R→ For the set of all real numbers.
• R⁺→ For the set of all positive real numbers.
• C→ For the set of all complex numbers.

Description of a Set

To make it more convenient for us. A set is often described in the following two forms. You can use one of these:

1. Roster form

In Roster form, A set is described by listing elements, separated by commas, within braces{ }.

Example:

1. The set of vowels of the English Alphabet may be described as {a, e, i, o, u}.
2. The set of even natural numbers can be described as {2, 4, 6, .....}.
3. If A is the set of all prime numbers less than 11, then A = {2, 3, 5, 7}.

2. Set-builder Form

In Set-builder form, A set is described by a characterising property P(x) of its elements x. In such a case the set is defined by {x : P(x) holds } or, {x | P(x) holds }, which is read as ' the set of all x such that P(x) holds'. The symbol ' | ' or ': ' is read as ' such that '.

Or

A set, a variable x(say) (to denote each element of the set ) is written inside the braces and then after putting a colon the common property P(x) possessed by each element of the set is written within the braces { }.

Examples:

1. The set E of all even natural numbers can be written as. E = {x : x is a natural and x = 2x for n ∈ N }
2. The set A = {1, 2, 3, 4, 5, 6, 7, 8} can be written as A = {x ∊ N: x ≤ 8}.
3. The set of all real numbers greater than -1 and less than 1 can be described as {x ∊ R : -1 ＜x＜1}.
4. The set A = { 0, 1, 4, 9, 16, ...} can be written as A = {x²: ∊ Z }.

Types of Sets

1. Empty Set

A set is said to be an empty set, null or void set if it has no element and is denoted by ф.

In the Roster method, ф is denoted by { }.

Examples:

1. {x ∊ R : x² = -2} = ф.
2. {x ∊ N : 5 < x < 6} = ф.
3. The set A is given by A = {x: x is an even prime number greater than 2} is an empty set because 2 is the only even prime number.

A set consisting of at least one element is called a non-empty or non-void set.

2. Singleton Set

A set consisting of a single element is called a singleton set.

Examples:

1. The set {5} is a singleton set.
2. The set {x : x ∊ N and x² = 9} is a singleton set equal to {3}.

3. Finite Set

A set is called a finite set if it is either a void set or its elements can be listed (counted, labelled) by natural numbers 1, 2, 3, ..... and the process of listing terminates at a certain natural number n (say).

Cardinal Number of A Finite Set

The number 'n' in the above definition is called the cardinal number or order of a finite set A and is denoted by n(A).

Examples:

1. A = {x : x ∊ Z and x² - 5x + 6 = 0} = {2, 3}
2. A = {x : x ∊ Z and x² = 36} = {6, -6}.
3. Set of even numbers less than 100.
4. Set of soldiers in the Indian army.

4. Infinite Set

A set whose elements cannot be listed as the natural numbers 1, 2, 3, ...., n, for any natural number n is called an infinite set.

Examples:

1. Set of all points in a plane.
2. Set of all lines in the plane.
3. {x ∊ R : 0 < x < 1}.
4. {x : x ∊ Z and x ＞ -10} = {-9, -8, -7, -6, -5, ......}

5. Equal Sets

Two sets A and B are considered equal if every element of A is a member of B and every element of B is a member of A.

Examples:

1. If A = {1, 2, 5, 6} and B = {5, 6, 2, 1} then A = B
2. A = {x : x - 5 = 0} = {5} and B = {x : x is an internal positive root of the equation x² - 2x - 15 = 0} = {5} then A = B.

Note: The elements of a set may be listed in any order.

6. Equivalent Sets

Two sets A and B are said to be equivalent if their cardinal numbers are the same or the same number of elements. i.e. n(A) = n(B).

The Cardinal Number means the number of elements or number of members in a given set.

Examples:

1. A = {0, a} and B = {1, 0} then n(A) = n(B)
2. P = {a, b, c} and Q = [1, 2, 3} then n(P) = n(Q)
3. X = {4, 2, 6, a, b} and Y = {3, 4, 5, 1, 2} then n(X) = n(Y)

Remember: Equal Sets can be equivalent sets but equivalent sets cannot be equal sets.

Subsets

Let A and B be two sets. If every element of A is an element of B, then A is called a subset of B. If A is a subset of B, we write A ⊆ B, read as "A is a subset of B" or "A is contained in B".

Thus,

⇒A ⊆ B 

iff

⇒a ∊ A and a ∊ B    form    A = { a }  and B = { a , b , c}

• If A is a subset of B, we say that B contains A or, B is a superset of A and we write B ⊃ A.
• If A is not a subset of B, we write A ⊈ B.
• Every set is a subset of itself. A ⊆ A
• The empty set is a subset of every set. A ⊆ B from A = ф and B = { 1, 2, 3} we can also say that an empty set is a proper subset (described below)  of a given set.
• A subset A of a set B is called a proper subset of B. If A ≠ B and we write A ⊂ B. In such a case, we can say that B is a superset of A. 
• A proper subset contains some but not all the elements of an original set. A ⊆ B from A = { 1, 2, 3} and B = {3, 4, 5} 
• An improper subset is a subset that contains every element of the original set. A ⊆ B from A = { 1, 2, 3} and B = {1, 2, 3, 4, 5}

Intervals as Subsets of R

In real line, various types of infinities subsets are designated as intervals which are defined below:

Closed Interval

Let a and b are two given real numbers such that a < b. Then, the set of all real numbers x such that a ≤ x ≤ b is called a closed interval and is denoted by [a, b].

Thus, 

[a, b] = [x ∊ R : a < x < b]

On the real line, [a, b] may be graphed as shown below.

Open Interval

Let a and b are two real numbers such that a < b, then the set of all real numbers x satisfying a < x < b is called an open interval and is denoted by (a, b) or ] a, b [.

Thus, 

On the real line, (a, b) may be graphed as shown below:

Encircling a and b means that a and b are not included in the set.

Examples,

1. (1, 2) = { x ∊ R: 1 < x < 2} is the set of all real numbers lying between 1 and 2 excluding the endpoints 1 and 2. This is an infinite subset of R.

Semi-open or semi-closed interval

If a and b are two real numbers such that a < b, then the sets (a, b ] = { x ∊ R : a ≤  x ≤ b } are known as semi-open or semi-closed intervals. ( a, b ] and [ a, b ) are denoted by ] a, b ] and [ a, b [ respectively.

Thus,

On the real line, these sets may be graphed as shown below:

The number b - a is called the length of any of the intervals ( a, b ), [ a, b ], [a, b ) and ( a, b ].

These notations provide an alternative way of designating the subsets of the set R of all real numbers.

Examples:

1. The interval [0, ∞) denotes the set R⁺ of all non-negative real numbers, while the interval (-∞, 0) denotes the set R^- of all the negative real numbers.
2. The interval (-∞, ∞) denotes the set R of all real numbers.

Universal Set

In set theory, there always happens to be a set that contains all sets under consideration i.e. it is a supper set of the given sets of each set. Such a set is called the Universal set and is denoted by ሀ.

Thus, a set that contains all sets in a given context. such a set is called the universal set.

Examples:

• When we study two-dimensional coordinate geometry, the set of all points in the xy-plane is the universal set.
• When we use sets containing a natural number, then N is the Universal set.
• When we use intervals on real lines, the set R of real numbers is taken as the universal set.
• If A = {1, 2, 3, } B = {2, 4, 5, 6} and C = {1, 3, 5, 7}, Then ⋃ = {1, 2, 3, 4, 5, 6, 7} can be taken as universal set.

Power Set

Let A be a set. then the collection or family of all subsets of A is called the power set of A and is denoted by P(A).

That is,

P(A) = {S : S ⊂ A }.

Since the empty set and the set A itself subsets of A therefore elements of P(A). Thus, the power set of a given set is always non-empty.

Examples:

• If A is the void set ф, Then, P(A) has just one element ф i.e P(ф) = (ф).
• Let A = {1, 2, 3}. Then, the subsets of A are : ф, { 1 }, [ 2 }, { 3 }, { 1, 2}, { 1, 3 }, { 2, 3 } and { 1, 2, 3}. Hence, P(A) = { ф, { 1 }, [ 2 }, { 3 }, { 1, 2}, { 1, 3 }, { 2, 3 }, { 1, 2, 3} }.
• If A = {a, { b } }, let B = { b } then the subsets of A = ф, { a }, { B } and { a, B }.  Hence, P(A) = { ф, { a }, { B }, { a, B } }.

Venn Diagrams

We know that sometimes pictures are helpful in our real object-oriented critical problem thinking. Swiss Mathematician Euler gives the first idea of representing a set by the points in a closed curve. After some time, British Mathematician John Venn (1834 - 1883) introduced this idea in the regular practice of solving set-related problems. So, it is known as the Venn Euler Diagram and is commonly also known as the Venn Diagram.

In the Venn diagram, the universal set ሀ is represented by points within the rectangle and its subset is represented by a closed curve ( usually a circle ) within the rectangle.

1. If a set A is a subset of a set B, then the circle representing A is drawn inside the representing circle B.
   
2. If A and B are not equal, they have some common elements. then to represent A and B we draw two intersecting circles.

   
3. Two disjoint sets are represented by two non-intersecting circles.

   

Operations on Sets

let's understand some operations on sets with the help of the Venn Diagram.

Important Theorem on the operations of Set theory.

Union Of Sets

Let A and B be two sets. The union of A and B is the set of all those elements which belong either to A or  B or to both A and B.

We use the notation A ∪ B ( read as " A union B" ) to denote the union of A and B.

Thus,

A ∪ B = { x : x ∈ A or x ∈ B }.

it means, x ∈ A ∪ B ⇔ x ∈ A or x ∈ B.

Similarly, x ∉ A ∪ B ⇔ x ∉ A or x ∉ B.

In the given figure, the shaded part represents A ∪ B. It is evident from the definition that A ⊆ A ∪ B, B ⊆ A ∪ B,

If A and B are two given sets such that A ∪ B, then A ∪ B = B. Also, A ∪ B = A if B ⊂ A.

Examples:

1. If A = { 1, 2, 3 } and B = { 1, 3, 5, 7 }, then A ∪ B = { 1, 2, 3, 5, 7 }.
2. Let A = { 1, 2, 3 }, B = { 3, 5 } and C = { 4, 7, 8 }. Then, A ∪ B ∪ C = { 1, 2, 3, 4, 5, 7, 8 }.
3. If A = { x : x = 2n + 1, n ∈ Z } and B = { x : x = 2n,  n ∈ Z }, then A ∪ B = { x : x is an odd integer } ∪ { x : x is an integer } = Z.

Intersection Of Sets

Let A and B be two sets. The intersection of A and B is the set of all the intersections of A and B.

Thus,

A ⋂ B = { x : x ∊ A and x ∊ B }

it means that, x ∊ A ⋂ B ⇔ x ∊ A and x ⋂ B

Evidently, A ⋂ B ⊆ A,  A ⋂ B ⊆ B.

If A and B are sets, A ⋂ B = A, if A ⊂ B and A ⋂ B = B, if B ⊂ A.

In the given figure, The shaded region represents A ⋂ B.

Examples:

1. If A = { 1, 2, 3, 4, 5} and B = { 1, 3, 9, 12 }, then A ⋂ B = { 1, 3 }
2. If A = { 1, 2, 3, 4, 5, 6, 7 }, B = { 2, 4, 6, 8, 10 } and C = { 4, 5, 6, 7, 8, 9, 10, 11 }, then A ⋂ B ⋂ C = { 4, 6 }

Disjoint  Sets

Two sets A and B are said to be disjoint, if A ⋂ B = ф.

If A ⋂ B = ф, A and B are considered intersecting or overlapping sets.

Examples:

1. If A = { 1, 2, 3, 4, 5, 6 }, B = { 7, 8, 9, 10, 11 } and C = { 6, 8, 10, 12, 14 }, then A and B are disjoint sets.

Difference Of Sets

Let A and B be two sets. The difference between A and B, written as A - B, is a set of all elements of A that do not belong to B.

Thus,

A - B = { x : x ∊ A and x ∉ B }

It means that, x ∊ A - B ⇔ x ∊ A and x ∉ B

In the given figure, the shaded part represents A -B.

Similarly,

the difference B - A is the set of all those elements of B that do not belong to A i.e. B - A = { x ∊ B : x ∉ A }.

In the given figure, the shaded part represents B - A.

Examples:

1. If A = { 1, 2, 3, 4, 5, 6, 7 } and B = { 3, 5, 6, 7, 9, 11, 13 }, then A - B = { 2, 4, 6 } and B - A = { 9, 11, 13 }.

Symmetric Difference Between Two Sets

Let A and B be two sets. The symmetric difference between sets A and B is the set ( A - B ) ⋃ ( B - A ), denoted by A ∆ B.

Thus,

A ∆ B = ( A - B ) ⋃ ( B - A ) = { x : x ∉ A ⋂ B }.

In the given figure, the shaded part represents A ∆ B.

Examples:

1. If A = { 1, 2, 3, 4, 5, 6, 7, 8 } and B = { 1, 3, 5, 6, 7, 8, 9 }, then A - B = { 2, 4 } and B - A = { 9 }. A ∆ B = { 2, 4, 9 }.

 Complement Of A Set

Let U be the universal set and let A be a set such that A U. Then, the complement of A concerning U is denoted by A' or A^c or U - A and is defined as the set of all these elements of U which are not in A.

Thus,

A' = {x ∊ U : x ∉ A },

It means that x ∊ A' ⇔ x ∉ A.

In the given figure, the shaded part represents A'

Examples:

1. Let the set of natural numbers N = { 1, 2, 3, 4, 5, 6, 7, 8,........} be the universal set and let A = {2, 4, 6, 8, ...........}, then A' = { 1, 3, 5, 7, ...........}.

Who introduced the set theory?

Georg Cantor (1845 - 1918) was a Russian mathematician who introduced the set theory.

How is set theory used in our daily life?

There are so many examples of sets theory in our daily life. Some famous examples where we can observe: Cricket players, Utensils are well organised in the Kitchen, Books are well organised in the Library, Arrangement of classes in school, analysing the data of business, learning SQL, etc.

Set Theory Symbols or Notations.

Symbol	Name	Explanation	Examples
{ }	Brace but pronounced as set	Within this bracket elements of the set present	A = { 1, 2, 3, 5 }
| or :	such that	This is used in the set to specify what is contained in the set	A = { x : x > 10 }
∊	Belong to or is a component of	this specifies that the element is a set member.	1 ∊ A where A = { 1, 2, 3 }
∉	not Belong to or is not a component of	this specifies that the element is not a member of a given set.	5 ∉ A where A = { 1, 2, 3 }
=	Equality relations	It defined sets that are equivalent in terms of the same components.	A = { 1, 2, 3 } and B = { 1, 2, 3 }, than A = B
≠	is not equal or Not Equality relation	It defined that sets are not equivalent in terms of the same components.	A = { 1, 2, 3 4, 5} and B = { 1, 2, 3 }, than A ≠ B
⊆	Subset	When all elements of A are present in B, A is a subset of B.	A = { 1, 2, 3} and B = { 1, 2, 3, 4, 5 }, then A ⊆ B
⊂	Proper subset	When some elements of A are present in B, then A is a proper subset of B.	A = { 1, 2, 3 } and B = { 1 , 3, 5 ,7 }, then A ⊂ B
⊈	is not a subset	When no elements of A are present in B, then A is not a subset of B.	A = { 9, 11, 13 } and B = { 1 , 3, 5 ,7 }, then A ⊈ B
⊇	Superset	When A contains all the elements of B, then A is a superset of B	A = { 1, 2, 3, 4, 5 } and B = { 1, 2, 3 }, then A ⊇ B
⊉	is not Superset	When A does not contain all the elements of B, then A is not a superset of B	A = { 1, 2, 3, 4, 5 } and B = { 1, 2, 7 }, then A ⊉ B
⋃	Union	Use to combine all the components of given sets.	A = { 1, 2, 3, 4 } and B = { 1, 2, 7 }, then A ሀ B = { 1, 2, 3, 4, 7 }
⋂	Intersection	Use to include the common components of given sets.	A = { 1, 2, 3, 4 } and B = { 1, 2, 7 }, then A ⋂ B ={ 1, 2 }
ф	Empty Set or pronounced as 'phi'	A set that does not contain any components or element	A = ф
U	Universal Set	A set that contains all the elements or components from relevant sets	A = { 1, 2, 3 } and B = { 2, 4, 6 }, then U = { 1, 2, 3, 4, 6 }
|A| or n{A}	The cardinality of a set	It refers to the number of items or components or elements in a particular set	If A = { 1, 2, 3, 4 }, then |A| or n{A} = 4
P(A)	Power Set	A Power set is a set of all subsets of set A, including itself and the null set.	A = { 1, 2, 3 }, then P(A) = { 1, 2, 3 } = { {ф}, {1},{2},{3},{1,2},{2,3},{1,3},{1,2,3} }
Ac or A'	Complement of a set	A complement set refers to all things not belonging to the provided set.	U = { 1, 2, 3, 4, 5, 6} and A = { 2, 4, 6}, then A' = U-A = { 1, 3, 5}

All the Important  Theorem and questions of Algebra on set theory. click here for more.

Conclusion  

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