Summary Ace Your College Algebra

Algebra I Summary

Amar Albourm

12/07/2022

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, Sets

0.1 Introduction
0.1.1 Definition
A set is an array of precisely defined and distinct elements, elements could be
any mathematical object: numbers, symbols, points in space, variables or even other
sets.
There are 3 types of sets:

1-Finite Sets: Arrays with a definite number of elements

S = {a, b, c, d}

2- Infinite Sets: Arrays with an indefinite number of elements

P = {2, 3, 5, 7, 11, ...}

3- Empty Set(Null Sets): A single array in which no elements are contained within
the set (Zero Entites), Denoted as:

E =∅
or

E ={}




0.1.2 Number Sets
N = {1, 2, 3, 4, ...} ⇒ Natural Numbers
Z = {0,
 ±1, ±2, ±3, ...} ⇒Integer Numbers
p
Q= |(p, q) ∈ Z, q ̸= 0 ⇒ Rational Numbers
q
R = {±x1 x2 x3 ...xn .y1 y2 y3 ...ym | 0 ≤ xi , yj ≤ 9} ⇒ Real Numbers
C = {z = a + bi | (Re(z) = a, Im(z) = b) ∈ R, i2 = −1} ⇒ Complex Numbers

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, 0.2 Working With Sets
0.2.1 Subsets
Let A be a subset of W, this means that every element of A is in W and we write:
A⊆W

Example
. C = {1, 6, 8} D = {1, 2, 3, 6, 8, 9}
C is a subset of D or:
C⊂D
Thus every element of C is in D, but C ̸= D


0.2.2 Improper Subset
Every set is a subset of itself: this is the Improper Subset

A⊆A

Note
The empty set is a subset of every set

∅⊆X

Disjoint Sets
Two sets are said to be disjoint sets if they have no element in common. Equiva-
lently, two disjoint sets are sets whose intersection is the empty set ∅.

A B




A∩B =∅

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