Set, Number system,discrete structure ppt.

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3. What is a Set?
 A set is a well defined collection of distinct objects. The objects that make up a set
(also known as the elements or members of a set) can be anything: numbers,
people, Animals, letters of the alphabet, other sets, and so on.
Example:
 A = {tiger, lion, puma, cheetah, leopard} (this is a set of large species of cats)
B = {a, b, c, ..., z} (this is a set consisting of the lowercase letters of the alphabet)
C = {1, 2, 3, ...} (this is a set of the numbers)

4. Types of Sets
1.Finite set
 A set which contains limited number of elements is called a finite set.
Example1. A = {1, 3, 5, 7, 9}.
 Here A is a set of five positive odd numbers less than 10. Since the number of
elements is limited, A is a finite set.
 A grade 5 class is a finite set, as the number of students is a fixed number.
2.Infinite set
 A set which contains unlimited number of elements is called an infinite set.
 The set of natural numbers N, is an infinite set as the counting of numbers does
not come to an end. For Example;
 N = {1, 2, 3, ……….} i.e. set of all natural numbers is an infinite set.
 W = {0, 1, 2, 3, ……..} i.e. set of all whole numbers is an infinite set.

5. 3.Singleton set
 A set which contains only one element is a singleton set.
 For Example; A= {set of even prime numbers}
Now A = {2}.
The only even prime number is 2. All other prime numbers are odd. Therefore A can
contain only one element, namely 2, so A is a singleton set.
4.Null set
 A Set which does not contain any element is called empty set or null set.
For Example; S = {x: x ∈Z, x = 1/n, n ∈ N}
N is natural number and Z is integer.
Since n is an integer, 1/n cannot be an integer. Therefore, S cannot contain an
element x which is an integer.
 Note:
 The Empty set is denoted as { } or by the greek letter Φ
 {{}} or {Φ} are not empty sets, because each contain one element, namely the
empty set Φ itself.

6. 5.Cardinal Number of a set or Cardinality of a set:
 The cardinality of a set is the number of elements a set contains. It is denoted as n
(A).
n (A) is read as the number of elements in set A
For Example;
 A = {1, 2, 3, 4, 5}
 The cardinality of set A is 5.
 It is denoted as n (A) = 5
6.Equivalent sets
 Two sets which have the same number of elements, i.e. same cardinality are
equivalent sets.
For Example;
P = {p. q. r, s, t} and Q = {a, e, i, o, u}
Since the two sets P and Q contain the same number of elements 5, therefore they
are equivalent sets.

7. 7.Equal sets
 Two sets that contain the same elements are called equal sets.
For Example; A = {1, 2, 3, 4, 5, 6, 7, 8, 9} and B = {1, 2, 3, 4, 5, 6, 7, 8, 9}
8.Overlapping sets
 Two sets that have at least one common element are called overlapping sets.
For Example; X = {1, 2, 3} and Y = {3, 4, 5}
The two sets X and Y have an element 3 in common. Therefore they are called
overlapping sets.
9.Disjoint sets
The two sets A and B should have no common elements , called disjoint sets.
For Example; C = {2, 4, 6} and D = {1, 3, 5}
The two sets C and D are disjoint sets as they do not have even one element in
common.

8. 10.Subset
 Set A is a subset of set B if every element of A is an element of set B.
If set A is a subset of set B, then it is denoted as A ⊂ B Let A
= {1, 2, 3} and B = {2, 3, 4, 1}
 Since every element of set A is present in set B too, we say A is a subset of B.
Note:
1. If two sets A and B are equal sets, then each one is a subset of the other.
If A = {a, e, i, o, u} and B = {vowels of English alphabets}, then A = B.
But, note that A ⊂ B and B ⊂ A.
Therefore, if A ⊂ B and B ⊂ A, then A = B
2. Every set is a subset of itself.
A ⊂ A
3. empty set is a subset of every set.

9. What are the Elements of set?
 The objects used to form a set are called its elements or its members.
Generally, the elements of a set are written inside a pair of curly braces
i.e { }. The name of the set is always written in capital letter.
For Example; A = {2, 4, 6, 8, 10, 12, 14}
Therefore: 2, 4, 6, 8, 10, 12, 14 are all elements of set A.
That is: 2, 4, 6, 8, 10, 12, 14 Є A (Where the symbol Є means ‘is an element of’).

10. What is Natural Number?
 The set of all natural numbers, normally is denoted by math symbol N. Natural
numbers happen naturally (hence the name) from counting objects.
For Example; The counting of objects in numbers 1, 2, 3 .... is known as natural
numbers.
N = {1, 2, 3, 4 ....}

11. What is Real Number?
 A real number is any element of the set R, which is the union of the set of rational
numbers and the set of irrational numbers.
For Examples;
 Natural numbers, whole numbers, integers, decimal numbers, rational numbers,
and irrational numbers are the examples of real numbers.
Natural Numbers = {1, 2, 3,...}
Whole Numbers = {0, 1, 2, 3,...}
Integers Z = {..., -2, -1, 0, 1, 2,...}
3/5, 10.3, 0.6, 12/5, 3/4, 3.46466466646666..., √ 2, √ 3 are few more examples.

12. What is Prime Number?
 A Prime Number can be divided evenly only by 1, or itself. And it must be a
whole number greater than 1.
For Example; 5 can only be divided evenly by 1 or 5, so it is a prime number, But 6
can be divided evenly by 1, 2, 3 and 6 so it is NOT a prime number (it is a
composite number).

13. What Is Rational Number?
 The Number in the form of p/q , where q ≠ 0 and bothe p & q are integers, called
rational numbers. Rational Number denoted by Q.
For Example; 1/3, 5/3,7/1, 8/3 etc.

14. What Is Empty Set?
 A Set which does not contain any element is called empty set or null set.
For Example; N is natural number and Z is integer.
Since n is an integer, 1/n cannot be an integer. Therefore, S cannot contain an
element x which is an integer.
 Note:
 The Empty set is denoted as { } or by the greek letter Φ
 {{}} or {Φ} are not empty sets, because each contain one element, namely the
empty set Φ itself.

15. What is Phi Symbol?
 Phi, Φ, 1.618…, has two properties that make it unique among all numbers.
 If you square Phi, you get a number exactly 1 greater than itself: 2.618…,
Or Φ² = Φ + 1.
 If you divide Phi into 1 to get its reciprocal, you get a number exactly 1 less than
itself: 0.618…,
Or 1 / Φ = Φ – 1.

16. What are Integers?
 Integers are a special group or category of numbers that:
 Consist of the set of numbers: {…-4, -3, -2, -1, 0, 1, 2, 3, 4…}
 All are positive and negative whole numbers, which do not include any fractional or
decimal part.
 It is denoted by Z (Z is for Zahlen , the German word for number)

17. What are whole numbers?
 The set of whole numbers include the natural numbers and 0. This set is denoted
by W
W = { 0,1,2,3,4,5,6,......}
In other words, whole numbers is the set of all counting numbers plus zero.
 Whole numbers are not fractions, not decimals.
 Whole numbers are non-negative integers.

18. What is intersection of set?
 Intersection of Sets is defined as the grouping up of the common elements of two
or more sets.
 It is denoted by the symbol ∩.
For Example;
 Sets A and B, A ∩ B = B ∩ A
 If Set A = {1, 2, 3, 7, 11, 13} and Set B = {1, 4, 7, 10, 13, 17},
A ∩ B is all the common elements of the set A and B.
Therefore, A ∩ B = {1, 7, 13}.
 If A = {5,6,7} and B = {2, 4, 6,8}
A ∩ B = {5, 6, 7} ∩ {2, 4, 6, 8} = {6} = {2, 4, 6, 8} ∩ {5, 6, 7} = B ∩ A.

19. What is union?
 Union of two sets A and B is obtained by combining all the members of the sets
and is represented as A ∪ B.
 In the union of sets, element is written only once even if they exist in both the sets.
For example;
 If A = {1, 2, 3, 4, 5} and B = {2, 4, 6}, then the union of these sets is
A ∪ B = {1, 2, 3, 4, 5, 6}
 If A = {5,6,7} and B = {2, 4, 6,8}
A ∪ B = {5, 6, 7} ∪ {2, 4, 6, 8} = {2, 4, 5, 6, 7, 8} = {2, 4, 6, 8} ∪ {5, 6, 7} = B ∪ A

20. is Zero positive or negative ?
 An integer is a whole number that can be either greater than 0,called positive, or
less than 0, called negative. Zero is neither positive nor negative.

21. Why we further divide numbers into sets?
 Suppose we need to divide the line into N groups such that the sums of the numbers of
each group are closest to the mean of these sums by some metric.
 we can choose to minimize sum of absolute differences, or variance, etc., depending o
For example;
 Numbers: 1 2 3 4 5 6 7 8 9 10, need to divide into 3 groups
 Let's say we want to minimize sum of absolute differences (SAD).
 Groups: (1) 1 2 3 4 5 6 (sum = 21); (2) 7 8 (sum = 15); (3) 9 10 (sum = 19)
 Mean = (21+15+19)/3 = 18.33, SAD = 21-18.33 + 18.33-15 + 19-18.33 = 6.67 <- That's
what we want to minimize.
 and it leads to the simplest solution.

22. Difference between rational and irrational number?
Rational Number
 A rational number is a number that can be written as a ratio. That means it can be written
as a fraction.
The number 8 is a rational number because it can be written as the fraction 8/1.
Likewise, 3/4 is a rational number because it can be written as a fraction.
Even a big fraction like 7,324,908/56,003,492 is rational, simply because
it can be written as a fraction.
Every whole number is a rational number, because any whole number can
be written as a fraction. For example, 4 can be written as 4/1, 65 can be
written as 65/1, and 3,867 can be written as 3,867/1.

23. Difference between rational and irrational number?
Irrational Numbers
All numbers that are not rational are considered irrational. An irrational number can be
written as a decimal, but not as a fraction.
An irrational number has endless non-repeating digits to the right of the decimal point.
Here are some irrational numbers:
π = 3.141592…
square root of 2 = 1.414213…
Although irrational numbers are not often used in daily life, they do exist on the
number line. In fact, between 0 and 1 on the number line, there are an infinite number
of irrational numbers!

24. Difference between natural numbers and real numbers?
The Natural Numbers
 The natural (or counting) numbers are 1,2,3,4,5,1,2,3,4,5, etc. There are infinitely many
natural numbers.
 The whole numbers are the natural numbers together with 0.
 The sum of any two natural numbers is also a natural number.
For example; (4+2000)=2004
 The product of any two natural numbers is a natural number (4×2000=8000). This is not true
for subtraction and division, though.

25. Difference between natural numbers and real numbers?
The Real Numbers
 The real numbers is the set of numbers containing all of the rational numbers and all of
the irrational numbers. The real numbers are “all the numbers” on the number
line. There are infinitely many real numbers just as there are infinitely many numbers in
each of the other sets of numbers. But, it can be proved that the infinity of the real
numbers is a bigger infinity.

26. AdvEntages of Empty set?
 The empty set is a set with no elements. We can use braces to show the empty set: { }.
Alternatively, this symbol, Ø, is often used to show the empty set.
 The Empty Set as a Solution Set
 If a problem has no solution, the solution can be represented by the empty set. For
instance, let's state this problem:
 'Name all the states in the United States that begin with the letter Z.'
 There are no states in the United States that begin with the letter Z. Therefore, the
solution to this problem is the empty set: Ø. However, we need to distinguish between
the empty set and the number zero as an answer. Let's reword the previous problem as
follows:
 'How many states in the United States begin with the letter Z?'

27. Advantages of Empty set?
 The answer to this question is 0. Using set notation, we would write the solution as
{0}. This solution contains one element, the number 0, so its cardinality is 1. It is
not empty. So we answer this problem in form of empty set { }
 The empty set as a sub set
 The empty set is also a subset of every set.

28. What is singleton set?
 Singleton set
 A set which contains only one element is a singleton set.
 For Example; A= {set of even prime numbers}
Now A = {2}.
The only even prime number is 2. All other prime numbers are odd. Therefore A can
contain only one element, namely 2, so A is a singleton set.

29. Zero is a rational number … why?
 To prove that “Zero” is a rational number or not.
We have to check if ‘0’ is a rational number or an irrational number. If we recall the
definition of rational numbers, they are the numbers which can be expressed in form
of p/q where p & q both are integers and q ≠ 0.
Moreover, we know 0 can be written as 0/1, we observe that both 0 & 1 are integers
and the denominator i.e. ‘1’ ≠ 0.
So we conclude that ‘0’ is a rational number and not irrational. We conclude that
all whole numbers, all Positive and Negative Numbers are rational numbers.

30. What are continue and discrete number?
 Continuous and discrete are mathematical terms.
Data can be Descriptive (like "high" or "fast") or Numerical (numbers).
And Numerical Data can be Discrete or Continuous:
Discrete data is counted,
Continuous data is measured
 Discrete Data
 Discrete Data can only take certain values.
For example,
 The number of students in a class (you can't have half a student).
 The results of rolling 2 dice can only have the values 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12

31.  Continuous Data
 Continuous Data can take any value (within a range)
For example
 A person's height: could be any value (within the range of human heights), not just
certain fixed heights,
 Time in a race: you could even measure it to fractions of a second,
 A dog's weight,
 The length of a leaf,

32. Difference between proper and improper set?
 Proper Subset:
 If A and B are two sets, then A is called the proper subset of B if A ⊆ B but B ⊇ A i.e., A
≠ B. The symbol ‘⊂’ is used to denote proper subset. Symbolically, we write A ⊂ B.
For example;
 1. A = {1, 2, 3, 4}
Here n(A) = 4
B = {1, 2, 3, 4, 5}
Here n(B) = 5
We observe that, all the elements of A are present in B but the element ‘5’ of B is not
present in A.
So, we say that A is a proper subset of B.
Symbolically, we write it as A ⊂ B

33. Difference between proper and improper set?
 Improper Subset
 An improper subset is a subset containing every element of the original set. A
proper subset contains some but not all of the elements of the original set. For example,
consider a set {1,2,3,4,5,6}. Then {1,2,4} and {1} are the proper subset while {1,2,3,4,5} is
an improper subset.

34. What is sub set?
Subset
 Set A is a subset of set B if every element of A is an element of set B.
If set A is a subset of set B, then it is denoted as A ⊂ B Let A = {1, 2,
3} and B = {2, 3, 4, 1}
 Since every element of set A is present in set B too, we say A is a subset of B.
Note:
1. If two sets A and B are equal sets, then each one is a subset of the other.
If A = {a, e, i, o, u} and B = {vowels of English alphabets}, then A = B.
But, note that A ⊂ B and B ⊂ A.
Therefore, if A ⊂ B and B ⊂ A, then A = B
2. Every set is a subset of itself.
A ⊂ A
3. empty set is a subset of every set.

35. What is notation?
 A system of symbols used to represent special things.
For example; In mathematical notation "∞" means "infinity".