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# Math Foundation for Data Science
This is my rework of topics covered in *Duke University's***Data Science Math Skills** course from Coursera, which is not in R or Python.
## Sets
A mathematical set is a collection of items, made of elements.
```{r Sets}
A = c(1,2,-3,7)
E = c('apple','pear','mango')
A
E
```
### Set Theory Notation
In set A, 2 is an element, and mango is an element in set E.
This is described as: 2 is an $\in$ A and mango is an $\in$ E.
### Cardinality
The cardinality of a set is simply the size or number of elements in it.
For set E, there are 3 items, so we write $|$E$|$= 3
# Intersection of sets
The intersection is the conditional argument symbol of sets, either "And" or "Or" regarding their elements.
```{r}
A = c(1,2-3,7)
B = c(2,8,-3,10)
D = c(5,10)
```
## Intersection "and"
The element or elements of 1 set have same value in the other set. Both values matching is required for condition to be true. *Note: The notation is to use { } for a set.*
Sets A and B both have 2 and -3 elements
A $\cap$ B = {2,-3}
Sets B and D do not both have shared elements, resulting in an empty set
B $\cap$ D = $\emptyset$
The full notation for this: A $\cap$ B = {x: X $\in$ and X $\in$ B}
## Intersection union
The union is not what you might assume, it is not the "and" but rather the **or** condition of elements in sets.
The elements in 1 or both sets to satisfy the condition.<br>
A $\cup$ B = {x: X $\in$ A or X $\in$ B} = {1,2,-3,7,8,10}
# Numbers
There are different types of numbers in math but the ones focused here are $\mathbb{R}$
and $\mathbb{Z}$ which are Real and Integers.
Take a negative 7 integer and get the absolute value.
```{r abs}
n = -7
abs(n)
```
## Numeric Conditionals
The values between variables compared on whether one variable is greater, less than or equal to, and greater or equal to the other value.
| symbol | meaning |
|--------| --------|
| > | greater than x |
| < | less than x |
| >= | greater or equal to x |
| =< | less than or equal to x |
**Boolean Logic** evaluates the condition and returns a True or False (1 or 0)
```{r}
a = 3.14
b = 6.5
x = 2
y = 17
# Boolean Logic
a > b
a <= b
# double equal signs means equality
# test for equality between both conditions
x*y == y*x
b < x
```
unknown value for z, check if condition is true.
```{r}
z = ?
z + 3 < 10
z < 10 - 3
z < 7
```
# Intervals
The intervals are notation of what elements are within the range of what is in a set.
## Closed set
A closed set has [] and means that x values are bound to the inner range <br>
[2, 3.1] = {X $\in$ : 2 < x < 3.1}
## Open set
The open set has ( ) and means values for x are within a range <br>
(5,8) = {X $\in$ : 5 < x < 8}
## Mixed set
The mixed set has ( ] notation and uses < and =< <br>
(-7.1, 15] = {X $\in$ : -7.1 < x =< 15 }
# **Summation** function $\sum$
The sum of values from i to x, then the x has a function performed on it.
This is testing the ``sum()`` function in R.
```{r}
sum(1:5)
sum(1:5, 9:19)
```
Testing out a for loop for the summation of $\sum_{i=0}^{5} {i^2}$