Week 1: Data Science Math Skills

  • Sets and What They’re Good For
  • The infinite World of Real Numbers
  • That Jagged S Symbol

Set

A set is a collection of things, in general set is a collection of elements

  • 1 ∈ A ( means 1 is an element of A)
  • 5 ∉ A ( means 5 is not an element of A)

Cardinality

Cardinality means the size of a set means the number of elements in it.

  • The cardinality of |A| = 6 (because there are 6 elements in A)
  • The cardinality of |B| = 3 (because there are 3 elements in B)

Intersections

Intersection simple means the elements which are present in both sets.The symbol for intersection is ∩ which is known as intersects and in general “and”

A ∩ C = ∅ (∅, is known as the empty set)

Unions

Unions simple means the elements which are present in either set.The symbol for union is ∪ which is known as union and in general “or”

Medical Testing Example

As a part of the course I also learned about a medical testing example. This medical example is best example to understand, how we can implement sets in real world. So, one think to note that VBS means “very bad syndrome” in this example.

S ∩ H = ∅ (No one person is both S and H)
S ∪ H = X ( either person have VBS or not)

Test

Now, lets talked about test

P ∩ N = ∅ (No one tests is both P and N)
P ∪ N = X (Either test is P or N)
  • S ∩ P are True Positive
  • H ∩ N are True Negative
  • S ∩ N are False Negative
  • H ∩ P are False Positive

Cardinality (size)

|S|
─── = proportion of people who have VBS
|X|
|H|
─── = proportion of people who do not have VBS
|X|
|S∩P|
───── = true positive rate
|X|
|H∩N|
───── = true negative rate
|X|
|H∩P|
───── = false positive rate
|X|
|S∩N|
───── = false negative rate
|X|

Venn Diagrams

Venn Diagrams is the way to visualize sets

  1. A = {1,5,10,2}
Fig 1.1 Single Set
Fig 1.2 Multiple Sets

Inclusion-Exclusion Formula

According to the formula,

|A ⋃ B| = |A| + |B|- |A ∩ B|
|A ⋃ B| = 8, {1,9,6,3,4,8} |A| = 5, {1,9,6,5,7}|B| = 5, {3,4,8,5,7}|A ∩ B| = 2, {5,7}|A ⋃ B| = |A| + |B|- |A ∩ B|8 = 5 + 5–28 = 10 -28 = 8
Fig 2.1 : Real Numbers

Absolute Value

For any x ∈ ℝ,

|x| = x, if x is non-negative|x| = -x, if x is negative

Inequalities

a < b “ a is less than b”a > b “a is greater than b”a ≤ b “a is less than or equal to b”a ≥ b “a is greater than or equal to b”a << b “a is much much lesser than b”

Closed intervals

In closed intervals square brackets [] are used ..

Open intervals

In open intervals parentheses () are used ..

Half-open intervals

In Half-open intervals both open and closed intervals are used ..

Sigma Notation

Sigma is represented by Σ, some example are:

i is just an “dummy indices”, that is used as a counter.

Distributive Property

Commutative Property

Summation of constants

Summation of constants means if the whole value is constant then sum that value to n times

Mean and Variance

To calculate the value of μ

The mean μz is also denoted by μ(z)
Formula for calculating mean
Formula for calculating variance

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