Review of Mathematics - Sets and Statistics / Probability

CAP 5507 – Module 1
Topic 1 – Math Review:
 Sets: A set is a collection of distinct items.
o Example: Consider the given collection of numbers: 1, 2, 3, 4, 10, 11,
12, 1, 2, 3, 11, 15, 5.
 In this collection, the set of numbers is {1, 2, 3, 4, 10, 11, 12,
15, 5}.
o Example: Consider the days of a week (Monday – Sunday).
 The set of days is {Monday, Tuesday, Wednesday, Thursday,
Friday, Saturday, Sunday}.
o Example: Consider the set of all integers from 1 through 100.
 The set of integers is {1, 2, 3,…,100}.
o Anything found in the set is considered to be an element.
 Empty Set: A set with no elements in it. This is denoted by using ∅ , which is
the same as using {}.
 Set Builder Notation: 1) R : Set of real numbers: (-∞ , ∞¿ . 2) Z : Set of all
integers: ({…., -3, -2, -1, 0, 1, 2, 3}). 3) N : Set of all natural numbers: ({0, 1,
2, 3}).
o Example: {x ∈ Z | x is even} = {…., -6, -4, -2, 0, 2, 4, 6, ….}.
o Example: {y ∈ R∨ y 2− y −6=0} = {-2, 3}, because
2
y − y −6=( y +2)( y−3).
 Notation Logistics:
o We start off with a set of brackets { }.
o This is then followed by the element variable name, such as x, y, etc.
o Use ∈ to denote that the element is a member of something.
o Following ∈, we have a set, so meaning the element is a member of a
set.
o We then use | to indicate “such that”.
o Lastly, we have a rule, which is like a specific constraint to follow.
 Example: {x ∈ R|x ≥2 }=¿
 Internal Notation:
o Open notation indicates that we do not have such element. Denote
using ().
o Closed notation indicates that we do have an element. Denote using [].

o Example: (2, 3) =

o Example: [-2, 4) =

o Example: [-1, 1] =
o Example: {x | x = n2 for n = 1, 2, 3, ….} = {1, 4, 9, 16, 25, ….}
 Set Operations: 1) ∩=¿ intersection. 2) ∪ = union. 3) ⊂ = subset.
o Intersection: x ∩ y = the set where the elements are found in x and y. x
∩ y = {x | x ∈ X | and x ∈ Y}.

,  Example: x = {1, 2, 3}, y = {3, 4, 7, 9} → x ∩ y = {3}.
 Example: x = {a, b, c}, y = {e, f, g} → x ∩ y = {}.
o Union: x ∪ y = the set where the elements are in at least one set. x ∪
y = = {x | x ∈ X | or x ∈ Y}.
 Example: x = {1, 2, 3}, y = {3, 4, 7, 9}→ x ∪ y = {1, 2, 3, 4, 7,
9}.
o Subset: x ⊂ y = every element of x is an element of y.
 Example: x = {1, 2, 3}, y = {1, 2, 3, 4, 5} → x⊂ y.
 Example: x = {1, 2, 3}, y = {1, 2, 4, 5} → x⊄ y.
 Vector: Obtained by putting elements in a different set together in a certain
order.
 Cartesian Product: Mathematical property that returns a set (specifically a
product set or product). We denote this by A x B, where A and B are sets:
o A x B = {(a, b) | a ∈ A and b ∈ B}.
o Example: A = {x, y, z}, B = {1, 2, 3} → A x B = {(x, 1), (x, 2), (x, 3),
(y, 1), (y, 2), (y, 3), (z, 1), (z, 2), (z, 3)}.
o Example: D = {M, T, W, R, F, Sa, Su}, T = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
11}.
 D x T = {(M, 1), (M, 2),…,(M, 11), (Tu, 1), (Tu, 2), …., (Tu, 11), ….
(Su, 1), (Su, 2), …., (Su, 11)}.
 Functions: Describes an association of elements of set x with other elements
from another set x. For each x ∈ X, the function will name a single element y
∈ Y to it. In this case, since the element y is associated with x, we have y =
f(x).
o Example: f(x) = x 2





o Notation - f: X→Y. This refers that the function maps x into y, where x is
the domain and y is the codomain.
 Example: f(x) = {(x, y) where x ∈ R , y ∈ R∨ y=x 2}, f: R → R
={f(x) | x∈ X } (range).
 Functions with Multiple Inputs: An example of this is f(x, y) = xy + 2y.
o Let x ∈ R and y ∈ R , therefore f: R x R → R
 Example: f(1, 3) = (1, 3) + 2(3) = 3 + 6 = 9