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title: "Math for Data Science"
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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 $\sum_{i=0}^{5} {i^2}$

```{r for loop}

library(stringr)

v = c(1:4)
for( i in v){
  x = i**2
  z = sum(x)
}
str_glue('The sum is {sum(x)}')

```



Sigma $\sum_{i=1}^{5} f(2i+3)$ 
```{r}

v = c(1:5)
for( i in v){
  k = i*2 + 3
  z = sum(k)
  print(z)
  h = sum(5, 7,9,11, 13)
}
str_glue('The sum is {h}')

```

Sum $\sum_{i=1}^{4} {3i^2}$

```{r}
v = c(1:4)
for( i in v){
  k = (3*i)**2
  z = sum(k)
}
str_glue('The sum is {z}')

```