Statements and Negations

Supplementary angle is up to

180 degree

Complementary angle is up to

90 degree

Counter example is :

to prove a statement is wrong

negation of statement

is the opposite of the statement

Compound statements types and definitions:

Conjunction:connect two or more statements with and p and q (sign is down)

Disjunction: connect two or more statements with (or) p\upsilonq

Conditional:if then statement (if p then q)

p\rightarrow q

p(hypothesis) and q(conclusion) cases:

the conjunction are all false except when both p and q are (TRUE)

the disjunction are all true except when both p and q are (FALSE)

the conditional are all True except when p is true and q is false

example to conditional:

If it is snowing then i will not stay home.

(thats a false statement)

Types of statements \rightarrow

Conditional Statement: use the given hypothesis and conclusion (p\rightarrowq)

Converse statement: exchange both hypothesis and conclusion (q\rightarrowp)

Inverse: negate both p and q of the conditional

Contrapositive: negate both p and q of the converse

Indirect proof:

to prove it the negation should be the opposite of the inequality

example:

if p<q then p\geq

and all should be the opposite to prove.

True or false

1)Points that lies on the same horizontal line have the same y coordinates

2)Points that lies on the same vertical line have different x coordinates

1)True

2)False, the have the same x coordinates nit different

Negation of conjunction vs Negation of disjunction

Negation of a conjunction (AND): becomes an (OR) and each part is negated.

Negation of a disjunction (OR):

becomes an (AND) and each part is negated.

Real numbers symbols

Natural numbers(N),1 to \infty

Whole numbers(W), 0 to \infty

Integers(Z), -\infty to \infty

Rational numbers(Q) p/q ,q\ne0

Irrational numbers(IQ), numbers that cannot be written in form of p/q

Converting decimal into a fraction steps:

First:

if the decimals is repeated for the same digit (10-1) for the denominator and take the digit for the numerator

example:

0.333.. : 10-1=9

3/9=1/3

if the decimal is repeated after two digits (100-1) for the denominator and the two digits for the numerator

example:

0.2727..: 100-1=99 , 27/99=3/11

Interval types

open circle means open interval

parentheses(,) means open interval

\le,\ge means open interval

closed circle means closed interval

brackets [,] means closed interval

<,> means closed interval

infinity are always opened (,) and there signs is \rightarrow,\larr

Functions vs One-to-One Functions

Functions different domains can have the same set of ranges, but same domain can’t have different set of domains

test on graph by VERTICAL LINE TEST.

One -to-One Function is the opposite of functions two different domains can’t have the same set of range

test on graph by HORIZONTAL LINE TEST.

What is the domain and range of the step function?

the domain is R

the range is Z

True or False

The monomial can have a negative integer exponent.

False, the monomials can only have positive integers exponents

perfect squares vs different of squares

perfect: (a±b)² = a± 2ab +b²

difference of squares: a² -b² (a-b)(a+b)

Complex numbers rules:

i^1 =i

i² =-1

i³ =-i

i^4 = 1

the rest repeats by using the remainder when dividing by 4

example:

i^15 = 15/4=3 remainder is 3

i³=-i

Quadratic,discriminant,and vertex formulas

Quadratic: x=b± root b² -4ac /2a

Vertex: x= -b/2a

(open up (minimum), open down(maximum)

Discriminant: b² - 4ac

to find the number of roots: >0 ,2 real solutions

<0, 2 complex solutions

=0, 1 real solution

Domains are:

domains are the zeros of the denominators

Prime polynomials are?

they are the polynomials that cannot be factored.

Remainder theorem

to find the remainder: take the side equation and equal to zero then substitute

to find the linear equation/ missing variable when remainder is given:try the choices

How do we find the number of roots?

if it was graph its how many times it intersects with x-axis (if it was on the x-axis but didn’t intersect its considered two roots)

if polynomial the highest degree

Sum and product of roots

Sum: -b/a

Product: c/a

Vertical vs Horizontal asymptotes

vertical are the zeros of the denominators

horizontal:degree of a> degree of b(no horizontal asymptote)

if degree of a<degree if b (y=0)

if degree of a =degree of b then (coefficient of a/coefficient of b)

Special case if 1/x (parent function)

1/x-h +k

if x=h f(x) is undefined

vertical asymptote is x=h

horizontal asymptote y=k

Types of Variations

Direct (as x increases, y increases and the opposite)

Inverse(as x increases, y decreases and the opposite)

Joint (y varies directly to two or more quantities)

Combined(y varies directly to z and inversely to x )

Variations formulas

Direct: y=kx ,k=y/x

y1/x1 =y2/x2

Inverse: y=k/x, k=xy

x1y1=x2y2

Joint: y=kyz, k=y/xz

y1/x1z1=y2/x2z2

Combined: xy=kz, k=xy/z

x1y1/z1=x2y2/z2