Set: Set is a well-defined list or collection of material objects such as books and pens or conceptual objects such as numbers and points

Types of Sets: The types of Sets which are described below are

1. Empty Set/Void set : A set having no element is called the empty set or null set or void set. It is denoted by ɸ(phi) or { } Example: P={ x:x is a male students in a girl’s college} is an empty set.
2. Singleton Set:A set having only one element in a set is called Singleton set E.g {0},{1}
3. Finite Set: A set containing a finite number of elements is known as a finite set. Example P={x:x is a vowel alphabets} and G={ x:x is a constant alphabet} are example of finite set
4. Infinite Set: A set which is not finite, is known as infinite set. Example: P={x:x is a negative integer} G={x:x is a prime number} are example of infinite set

Relation between Sets: The Relation different the sets are

a. Subset: A set A is said to be a subset of the set B if every element of A is also an element of B. It is denoted by A⊇  B. This is read as A is contained in B or B contains A. Here B is also known as super-set of A and we write as  B ⊇ A

                                      If every element of set A is also an element of set B but there is at least one element of B which is not an element of set A then, A is know as the proper subset of B. It is denoted by A ⊂ B

Proper Subset Formula: If a set has “n” items, the number of subsets for the supplied set is 2ⁿ, and the number of appropriate subsets of the provided subset is computed using the formula 2ⁿ – 1.

Example:

      A={x:x is a Natural Number }

      B={x:x is an Even Number}

Then, B ⊂A

Important points on Subsets:

i) If a set contains n elements then we can form 2ⁿ  number of subsets of that set.

ii) Out of all subsets of any set one is empty and one is improper

iii) Proper subsets of a set= total subsets – improper subsets = 2ⁿ – 1.

iv) Non-empty proper subsets= total subsets – empty sets – improper subsets= 2ⁿ – 1-1 = 2ⁿ – 2.

Example: How many subsets and proper subsets does a set have?

If set A has the elements, A = {p,q}, then the proper subset of the given subset are { }, {p}, and {q}.

Here, the number of elements in the set is 2. 

We know that the formula to calculate the number of proper subsets is 2ⁿ – 1. 

= 2² – 1

= 4 – 1

= 3

Thus, the number of proper subset for the given set is 3 ({ }, {p}, {q}).

b. Equal Sets: Two sets A and B are said to be equal or identical or same if they have the same elements. They are denoted by A=B.

Thus, if A⊆ B and B⊆ A, then A=B.

Also if If x∈A implies x∈B and x∈B implies x∈A, then A=B.

Example:

A={a,e,i,o,u} and B={x:x is a vowel}

Then, A=B

c. Intersecting sets: Two sets A and B are said to be intersecting if they have atleast one element in common

Example: If P={g,u,p,t,a} G={t,u,o,r,i,a,l} then P and G are intersecting sets because the element {u,t,a} are common

d. Disjoint sets: Two sets A and B are said to be disjoint if they have no element in common

Example: P={x:x is a constant alphabet} and G={x:x is a vowel alphabet } are disjoint sets because there is no element common between A and B

e. Power set: The collection or the set of all possible subsets of any set S is called the power set of S. It is denoted by P(S) or 2^S.

Example: If S={1,2} then it’s subsets are ɸ, {1}, {2}, {1,2}.

So, the power set of S is 2^S= 2^2 = 4 = { ɸ, {1}, {2}, {1,2} }

f. Equivalence sets: Two sets A and B are said to be equivalent if they have the same number of elements. They are denoted by A~B.

Example: A={p,r,a,v,i,n} and B={a,e,i,o,u} are the equivalent set because they have same number of elements.

Operation on Sets: The following are the operations on sets.

a. Union: The union of two sets A and B is defined as the set of all elements which belongs to A or B or both. It is denoted by AUB and read as A union B or A cup B.

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

then, AuB={1,2,3,4,5,6} U {2,4,6,7}={1,2,3,4,5,6,7}

The Venn-diagram of A U B is given below:

b. Intersection: The intersection of two sets A and B is the set of all elements which belonging to both sets A and B. It is denoted by AnB and read as A intersection B or A cap B.

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

then, AnB={1,2,3,4,5,6} n {2,4,6,7}={2,4,6}

The Venn-diagram of A n B is given below:

c. Difference: The difference of two sets A and B is the set of all elements of A but not belonging to B. We denotes it by A-B and reads as A difference B.

Symbolically, A-B={ x:x∈A and x∉B}

similarly, B-A={ x:x∈B and x∉A}

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

then, A-B={1,2,3,4,5,6}-{2,4,6,7}={1,3,5}

and B-A={2,4,6,7} – {1,2,3,4,5,6} ={7}

The Venn-diagram of A-B and B-A is given below:

d. Complement: The complement of a set A is the set of all elements in the universal set U that do not belong to A. It is denoted by Ā and read as A bar.It’s symbols is A’ and A^c

Symbolically, Ā = { x:x∈U and x∉A} = { x:x∉A }

Example: U={1,2,3,4,5,6,7,8,9} and A={2,4,6,8}

Ā=U-A={1,2,3,4,5,6,7,8,9} – {2,4,6,8} ={1,3,5,7,9}

The Venn diagram of the complement of a set is given below:

e. Symmetric Difference: The union of the differences A-B and B-A of two sets A and B is called the symmetric difference of A and B.It is denoted by A△B and reads as A delta B.

Symbolically, A△B =(A-B) U (B-A)

={x:x∈A-B or x∈B-A}

Thus, x∈A△B => x∈A or x∈B but x∉AnB.

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

then, A-B={1,2,3,4,5,6}-{2,4,6,7}={1,3,5}

and B-A={2,4,6,7} – {1,2,3,4,5,6} ={7}

then, A△B =(A-B) U (B-A) = {1,3,5} U {7} = { 1,3,5,7}

The Venn diagram of A△B is given below:

Cardinal Number of a finite set: The total number of elements in any set is know as the cardinal number of the set. It i denoted by n(A).

Two Sets A and B are said to have the same cardinality or cardinal number if they are equivalent(i.e there is a one-to-one correspondence between them)

Example: If A={g,u,p,t,a} then n(A)=5(i.e counting the number of element in a set)

Laws of Algebra of Sets

1. Properties of inclusion and Equality Relations: Let A,B and C be subsets of a universal set U.Then
   • A⊆ B, B⊆ C => A⊆ C
   • A=B => B=A
   • A=B, B=C => A=C
   • A⊆ɸ => A=ɸ
2. Properties of Unions:Let A,B and C be subsets of a universal set U.Then
   • AuA=A
   • Auɸ= ɸ
   • AuU=U
   • AuB=ɸ =>A=ɸ and B=ɸ
   • AuB=BuA
   • (AuB)uC=Au(BuC)
3. Properties of Intersections:Let A,B and C be subsets of a universal set U.Then
   • AnA=A
   • Anɸ= ɸ
   • AnU=A
   • AnB=BnA
   • (AnB)nC=An(BnC)
4. Miscellaneous Properties: Let A,B and C be subsets of a universal set U.Then
   • Au(BnC)=(AuB) n (AuC)
   • An(BuC)=(AnB) u (AnC)
   • Au Ā=U
   • AnĀ=ɸ
   • (Ā)’=A
   • A-(BnC)=(A-B)u(A-C)
   • A-(BuC)=(A-B)n(A-C)
   • An(B-C)=(AnB)-(AnC)

5. De-Morgan’s Law;

Let A,B and C be subsets of a universal set U.

1. Inclusion Laws:
   • A⊆ A
   • A⊆ B, B⊆ A => A=B
   • A⊆ B,B⊆ C => A⊆ C
   • ɸ⊆ A
2. Identity Laws
   • Auɸ=A
   • Anɸ=ɸ
   • AuU=U
   • AnU=A
3. Idempotent Laws
   • AuA=A
   • AnA=A
4. Commutative Laws
   • AuB=BuA
   • AnB=BnA
5. Associative Laws
   • Au(BuC)=(AuB)uC
   • An(BnC)=(AnB)nC
6. Distributive Laws
   • Au(BnC)=(AuB)n(AuC)
   • An(BuC)=(AnB)u(AnC)
7. Complement Laws
   • Au Ā=U
   • AnĀ=ɸ
   • (Ā)’=A
   • ɸ’ = U
   • U’=ɸ
8. De-Morgan’s Law;
   • 
9. Difference laws;
   • A-(BuC)=(A-B)n(A-C)
   • A-(BnC)=(A-B)u(A-C)

Real Number System

Natural Numbers: The simplest and the most familiar unending chain of consecutive numbers 1,2,3,…………… are known as the natural numbers.The natural numbers are also known as the counting numbers. The set of natural numbers are denoted by N.

N={1,2,3,4,……………………….}

Example: 10,15 ∈ N then 10+15=25 ∈ N and 10 x 15=150 ∈ N

Integers: The set of natural numbers together with their negatives including zero are known as the set of integers. 1,2,3,4,5,……………………. are known as positive integers and -1,-2, -3, -4, -5 ,………….. are known as negative integers. The set of integers are denoted by Z or I.

Z={………… -3, -2, -1, 0 ,1, 2, 3, 4, ………………….}

Integers may be either negative values,positive or 0

Rational Number: A number in the form of p/q where p and q are integers and q≠0 is known as a rational number.It is denoted by Q.

Q={x:x=p/q, p and q are integers and q≠0}

A rational number can be expressed as a terminating decimal or a repeating decimals.

Example: 5, -1, -6/7,1/9, 0.54, 0.3231,……….. etc are the rational numbers.

Irrational Number: Numbers which are not rational are know as irrational. A number which cannot be expressed in the form of p/q where p and q are integers and q≠0, is known as an irrational number. It is denoted by Ǭ(complement of Q)

Example: √8 , Π, e, ∛7 are the example of irrational numbers.

Real Numbers: The set of rational and irrational numbers taken together, form a new system of numbers known as the real number system. The union of the set of rational and irrational numbers is known as the set of real numbers. So, by the set of real numbers,we have the set of natural numbers, the set of integers, the set of rational numbers and the set of irrational numbers. It is denoted by R.

Interval: Let a and b be two numbers on the real lines.Then the set of points on the real line between a and b is known as an interval. a and b are known as the end point of the interval. It is denoted by I.

The following are the types of intervals

1. Open-interval: An interval not containing the end points a and b is known as an open interval.It is denoted by (a,b)

Symbolically, (a,b)={x:a<x<b}

The graph of the open interval(a,b) is shown below

2. Closed-interval: An interval containing both the end points a and b is known as an closed interval.It is denoted by [a,b]

Symbolically, [a,b]={x:a≤x≤b}

The graph of the closed interval [a,b] is shown below

3. Left Open Interval: An Interval not containing the end point a and containing the end point b is known as a left open interval. It is denoted by (a,b]

Symbolically, (a,b]={x:a<x≤b}

The graph of the left open interval(a,b] is shown below

4. Right Open Interval: An Interval containing the end point a and not containing the end point b is known as a right open interval. It is denoted by [a,b)

Symbolically, [a,b)={x:a≤x<b}

The graph of the right open interval [a,b) is shown below

Absolute Value: The distance of a number from zero on the number line, regardless of direction. In simpler terms, it’s the magnitude of a number without considering whether it’s positive or negative.It is denoted by |x|

• |x| = +x for x > 0 
• |x| = -x for x < 0

Example: |-5| = -(-5)=5, |9|= +(9)=9

|x|<a => -a<x<a => x∈a(-a,a).

The properties of absolute values are;

1. Let x be any real number.Then,
   • |x| ≥0
   • |x| ≥ x and |x| ≥ -x
   • -|x| ≤ x ≤ |x|
2. For any two real numbers x and y,
   • |x+y|≤|x|+|y| (Triangle Inequality)
   • |x-y|≥|x|-|y|
3. For any two real numbers x and y,
   • |xy|=|x||y|
   • |x/y|=|x| / |y| , y≠0

Logic:Logic is the process by which we arrive at a conclusion from the given statement with a valid reason.Logic tells us the truth and the falsity of the particular statement

Statements:An assertion expressed in words or symbols, which is either true or false but not both at the same times, is known as a statement.

Example;

i. water is essential for human

ii. 3+7=10

iii. A triangle has four sides

iv. Open the door

v. How are you?

(i),(ii) and (iii) are statements as (i) & (ii) are true but (iii) is false but (iv) & (v) are not statement because they do not declare the truth or falsity

Logical Connectives: Compound statements are made from the simple statements by using the words or phrases like “and”, “or”, “if……then” and “if and only if” and they are known as logical connectives or simply connectives.

1. Conjunction: Two simple statements combined by the words “and”(or equivalent word) to form a compound statement is known as conjunction of the given statement. It is denoted by ^.

If p and q are two simple statements, then their conjunction is symbolized by p^q.In case of conjunction of p and q, if p is true and q is true then p^q is true.Otherwise p^q is false.

The truth table of the conjunction of the statements p and q is presented below:

2. Dis-junction: Two simple statements combined by the words “or”(or an equivalent word) to form a compound statement is known as dis-junction of the given statement. It is denoted by v.

If p and q are two simple statements, then their dis-junction is symbolized by pvq. In case of dis-junction of p and q, if p is true or q is true or both p and q are true then pvq is true.Otherwise pvq is false.

The truth table of the dis-junction of the statements p and q is presented below:

3. Negation: A statement which denies the given statement is known as the negation of a given statement. The symbol used for the negation is ∼. If p is the given statement, then its negation is symbolized by ∼p.

The truth table of the negation of the statements p is presented below:

4. Tautology: A compound statement which is always true, whatever may be truth values of its components, is known as tautology

5. Contradiction: A compound statement which is always false, whatever may be truth values of its components, is known as contradiction.

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