Numbers and Sets

Question 1

What are arithmetic operations?

Answer 1

Arithmetic Operations : + , -, x , / , M^n
a mod b = c ==> a when divided by b gives remainder c
Divisibility - a|b => a divides b , e.g: 2|6

Question 2

What are various types of numbers?

Answer 2

N: Natural Numbers {0,1,2,… }
Z: Integers {… -3,-2,-1,0,1,2,3,…. }
Q: Rational Numbers { 2/3,4/6,8/23 ,… }

Question 3

What are factors?

Answer 3

If a|b then a is called a factor of b

Question 4

What are prime numbers?

Answer 4

If the factors of a number ‘x’ are 1 and x itself, x is called a prime number. Prime numbers have only 2 factors. Hence, 1 is not prime and 2 is the only even prime number.

Question 5

What is prime factorization?

Answer 5

Any number can be uniquely represented as the product of its prime factors. converting a number into this representation is called prime factorization.

Question 6

Is there a largest price number?

Answer 6

No. The set of prime numbers is infinite

Question 7

What are rational numbers?

Answer 7

Numbers of the form p/q. where p,q are integers. The representation of integers in rational format is not unique. one number could have multiple representations (eg: 2/4 = 4/8=8/16 etc). We could get the rational number to the reduced for by getting their greatest common divisor to 1 i.e., gcd(p,q)=1 i.e, remove all the common factors between them.
Unlike Integers which are discrete ( definite next and previous numbers possible). Rational numbers are dense, there is no definite next or previous number)

Question 8

What are irrational numbers and real numbers?

Answer 8

Irrational numbers cannot be written in the form p/q. eg.: sqrt of all numbers which are not perfect squares.

Real numbers = set of all rational and irrational numbers.
Real numbers extend rational numbers.
Real numbers are dense, like rationals
Every natural number is an integer
Every integer is a rational number
Every rational number is a real number
Complex numbers extend real numbers

Question 9

What are sets?

Answer 9

A set is a collection of items. Set may be infinite. Sets need not have members of a uniform type. Sets are unordered. Duplicates don’t matter. items in a set are called elements. Sets are best represented diagrammatically by Venn Diagrams.

Sets can also contain other sets.

Question 10

What is cardinality?

Answer 10

Cardinality of a set is the number of items in the set.
Cardinality could be finite or infinite.
finite sets can be listed out explicitly within {}

Question 11

Is every collection of items a set?

Answer 11

Not every collection of items is a set. Collection of all sets is not a set.
Russel’s paradox.

Question 12

What is a subset?

Answer 12

X is subset of Y if every element in X is also an element of Y.
Note: every set is a subset of itself. If set has ‘n’ elements then there is a total of 2^n possible subsets.

if X is subset of Y but X != Y then X is proper subset of Y

Question 13

How to establish equality of sets?

Answer 13

If X is subset of Y and Y is a subset of X the it implies that X=Y

Question 14

What are some basic sets?

Answer 14

The empty set has no elements (empty set is a subset of all sets)

powerset - set of subsets of a set.

Note: ø means set with no elements where as {ø} means a set with one empty element. Cardinality of ø is zero, where as cardinality of {ø} =1 (including the empty element)

powerset of ø = {ø}

Question 15

What is set comprehension?

Answer 15

Set Comprehension is a way to define a set from a different known set.
eg: {x | xє Z , x mod 2=0} =>represents set of All even integers

Question 16

What are types of intervals?

Answer 16

1. open interval (0,1) => excludes both 0 and 1
   1a. left open interval (0,1] => excludes 0 but includes 1
   1b. right open interval [0,1) => includes 0 but excludes 1
2. Closed interval [0,1] => includes both 0 and 1

Question 17

What are set operations?

Answer 17

1. union - combines two sets (excludes duplicates)
2. Intersection - common elements from both the sets
3. difference = elements in first set and not in second set.
4. complementation = elements not in a particular set (calculated on a larger universal set)

Question 18

What are relations?

Answer 18

Cartesian product : A x B = { (a,b)| a E A, bE B }
combine each and every element from A to each and every element from B
Note: order matters , (0,1) != (1,0)

2 sets could be visualized as cordinate axis. relations could be visualized as graphs.

Relation is subset of cartesian product which satisfies a filter or condition. Relation could be binary (2 axis), tertiary(3 axis) etc. Notation: (a,b)ER (or) aRb

Question 19

What are some of the basic relations?

Answer 19

1. Identity relation I = {(a,b) |(a,b)E A x A, a=b} or I={(a,a)|aEA}
2. Reflexive relations => every element is related to itself. i.e., (a,a) is in relation. Even if an element is not present in the relation, the relation is not reflexive. all elements must be present and be related to itself. (could have additional non identical pairs as well) - informally defined as loops
3. Symmetric relations => if a is related to b then b is also related to a (not all elements are required.) - informally defined as connections
4. Anti -symmetric relations=> if a is related to b then b is NOT related to a. (not all elements required) - Informally defined as one way connections only
5. Transitive relations => if a is related to b, and b is related to c. then a is related to c (not all elements are required but should satisfy with all points in the relation) - informally defined as shortcuts

Question 20

What are equivalence relations?

Answer 20

A equivalence relation is reflexive, symmetric and transitive. typically partitions the set into partitions of disjoint sets. and each partition is called equivalence class.

Question 21

Why are relations important?

Answer 21

A Table of data could be thought of as a relation. Each column could be imagined to be a set. If the table has n columns then each row could be thought of as a nary relation element.

Question 22

What is a function?

Answer 22

A function is a rule to map inputs(domain) to outputs (co-domain). The input is called a parameter. Range is the set of actual values that the output can take (could be bounded or unbounded)
A relation (Rf) (or more?) could be associated with each function.
Properties of Relation (Rf):
1. Defined on entire domain
2. Single Valued

Question 23

What are some of the properties of functions?

Answer 23

1. Injective (one to one) - Every unique input will have unique output
2. Surjective (onto) - Every unique output is mapped to some input
3. Bijection (one to one and onto) - every unique input maps to unique output.

Question 24

What is a countable set?

Answer 24

If a bijection exists between Natural numbers set (N) ( which is an infinite set) and the test set (S) then we say S is a countable set.

eg: Integers set (Z) is countable
Rational numbers (Q) is also countable even though Q is dense.
Real Numbers (R) is NOT countable.

Question 25

How to prove if a set is uncountable?

Answer 25

Use diagonlization argument by Georg Cantor. enumerate all possible sequences of enumerations of the test set (S). If we flip the bits of the diagonal elements, we get a new sequence which is not in the list already. Hence, what ever be the enumerations we provide, we cannot conclusively prove that one more sequence cannot be made. Hence, we cannot have complete enumeration of all possibilities of the test set (S) .

Question 26

What are important points in a function?

Answer 26

Maxima (local) - Maximum value within the bounded interval
Minima (local) - Minimum value within the bounded interval

Maxima (Global) - overall maximum value of the function
Minima (Global ) - Overall minimum value of the function.

Root - For what value of input the value of output is zero.

Question 27

how to compare functions?

Answer 27

Asymptotic analysis

Question 28

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