Math Final Definitions CH 1 - 12

set equality

2 conditions: Every element of A is an element of B; and; Every element of B is an element of A

skewed to the left 12.2

a non-symmetric distribution with a tail extending out to the left shaped like a J

vertical angles

a pair of angles that extend to form another angle vertical angles that have equal measures

rectangle

a parallelogram with a right angle (and consequently, four right angles)

rhombus

a parallelogram with all sides having equal length

trapezoid

a quadrilateral with one pair of parallel sides

fallacy

an invalid argument arguments exhibiting illogical reasoning

Connective Symbol Type of Statement

and ∧ Conjunction or ∨ Disjunction not ∼ Negation

obtuse angle

angles that measure more than 90° but less than 180°

perpendicular bisect

any chord of a circle passes though the center of the circle

experiment

any observation, or measurement of random phenomenon

normal distribution 12.5

any random variable whose graph has this characteristic shape (bell-shaped)

event

any subset of a sample space

premises

assumptions, laws rules, widely held ideas, or observations

variance

average that results is itself a measure of dispersion

standard deviation 12.3 Pg 672

based on deviations from the mean of the data values >>> Let a sample of n numbers x1,x2,...,xn have mean x¯. Then the sample standard deviation, s, of the numbers is calculated as follows. s=∑(x−x¯¯)2n−1−−−−−−−−−−−√ The individual steps involved in this calculation follow. Step 1 Calculate x¯,the mean of the numbers. Step 2 Find the deviations from the mean. Step 3 Square each deviation. Step 4 Sum the squared deviations. Step 5 Divide the sum in Step 4 by n−1. Step 6 Take the square root of the quotient in Step 5.

rectangular parallelepiped

box

semicircle

two parts of equal size of a circle

right circular cylinder

typical tin can is example

two-part task

use product table

quantifiers

used to indicate how many members in a group being considered exhibit a particular property or characteristic:

expected (theoretical) frequencies 12.1

uses binomial probability formula

symbolic logic

uses letters to represent statements, and symbols for words such as and, or, not.

skewness coefficient 12.3

uses to measure the degree of skewness involves both central tendency and dispersion and is calculated

set-builder notation

uses variables to define elements Example: { x | x is a vowel}

continuous random variable 12.5

variable whose values are not restricted

product table

vertical and horizontal table of values

In order to determine the amount of liquid a spherical tank will​ hold, would you need to use volume or surface​ area? Choose the correct answer below.

volume

perpendicular lines

when 2 lines inter sect to form right angles

transversal line

when a line intersects 2 parallel lines

binomial Probability distribution

when outcomes of an experiment are divided into just 2 categories, success and failure, the associated probabilities are "binomial" Bernoulli trials after James Bernoulli

infinite set

whenever a set is so large that its cardinal number is not found among the whole numbers

written method

writing out explanation of the elements Example: "the set of vowels in the English alphabet"

one degree

written 1°, comma is defined to be 1/360 of a complete rotation

The box has a volume of 1575. . Find x. width= 14 length= 15 height= x

x = 7.5

List all the elements of the following set. Use set notation and the listing method to describe the set. {x|x is an odd integer between -8 and 5

{-7, -5, -3, -1, 1, 3}

Integers

{..., -3, 2, 1, 0, 1, 2, 3, ... }

List all the elements of the following set. Use set notation and the listing method to describe the set. The set of all whole numbers not greater than 1

{0, 1}

Whole numbers

{0,1,2,3,4,.....}

List all the elements of the following set. Use set notation and the listing method to describe the set. The set of all counting numbers less than or equal to 6

{1, 2, 3, 4, 5, 6}

List all the elements of the following set. Use set notation and the listing method to describe the set. {x |x| is an odd whole number less than 11}

{1, 3, 5, 7, 9}

List all the elements of the following set. Use set notation and the listing method to describe the set. {6,7, 8, ..., 14}

{6, 7, 8, 9, 10, 11, 12, 13, 14}

List all the elements of the following set. Use set notation and the listing method to describe the set. The set of all whole numbers greater than 6 ad less than 15.

{7, 8, 9, 10, 11, 12, 13, 14}

Rational numbers

{p/q | p and q are integerds and q~0}

Rational numbers

{p/q | p and q are integers and q~0}

Real Numbers

{x|x is a number that can be expressed as a decimal}

Irrational numbers

{x|x is a real number and x cannot be expressed as a quotient o integers}

Find ​(a) the volume and ​(b) the surface area of the figure. A rectangular box has length 9 inches, width 6 inches, and height 2.7 inches.

​(a) Volume= 145.8 in. cubed ​​(b) Surface area equals 189 in. squared

palindromic numbers

#'s that are palindromes

Find the measure of each marked angle. Assume the lines are parallel.

(2x - 24)` = 34` (x+5)` = 34`

Proof of Formulas

(a) P(A and B)=P(A)⋅P(B|A) (b) Therefore, P(B|A)=P(A and B)P(A). (c) Therefore, P(B|A)=n(A and B)/n(S)n(A)/n(S). (d) Therefore, P(B|A)=n(A and B)n(A).

permutation

(arrangements) Permutations involve the number of arrangements of n things taken r at a time, where repetitions are not allowed.; >> nPr >> must satisfy these conditions: 1. repetitions are not allowed; 2. order is important

law of large numbers

(law of averages)A theoretical probability really says nothing about one, or even a few, repetitions of an experiment, but only about the proportion of successes we would expect over the long run. >>> As an experiment is repeated more and more times, the proportion of outcomes favorable to any particular eveny will tend to come closer and closer to the theoretical probability of that event

connectives

(logical connectives) such as *and, or, not, if,... then, can be used in forming compound statements

expected value

(mathematical expectation) the quantity expected If a random variable x can have any of the values x1,x2,x3,...,xn, and the corresponding probabilities of these values occurring are P(x1),P(x2),P(x3),...,P(xn), then E(x), the expected value of x , is calculated as follows. E(x)=x1⋅P(x1)+x2⋅P(x2)+x3⋅P(x3) + ⋯+xn⋅P(xn)

combination

(subset) the number of size-r subsets, given a set of size n—is written nCr; There are n things available and we are choosing r of them, so we can read nCr as "n choose r"

truth value

(the truth or falsity) of statements with multiple parts. The truth value of such statements depends on their components.

universal qualifiers

*indicate all members* all, each, every, no(ne) be careful when forming the negation of a statement involving quantifiers

Zero factorial

0! = 1

Factorials Table 0 - 10

0! = 1 1! = 1 2! = 2 3! = 6 4! = 24 5! = 120 6! = 720 7! = 5040 8! = 40,320 9! = 362,880 10! = 3,628,800

outlier 12.2

1) A piece of data that is quite separated from the rest of the data 2) should usually be considered as possible errors in the data 3) A value much greater or much less than the others in a data set

class width 12.1

1) for the distribution is the difference of any two successive lower class limits 2) or of any two successive upper class limits

weighting factor 12.2

1) the number of units in 2) a mathematical factor used to make a disproportionate sample representative.

grouped frequency distribution 12.1

1. Make sure each data item will fit into one, and only one, class. 2. Try to make all classes the same width. 3. Make sure the classes do not overlap. 4. Use from 5 to 12 classes. Too few or too many classes can obscure the tendencies in the data.

modus ponens

> >>law of detachment

decile 12.4

>> are the nine values (denoted D1, D2,..., D9) along the scale that divide a data set into ten (approximately) equal-sized parts >> We can evaluate deciles by finding their equivalent percentiles.

circle

>>a simple closed curve defined as follows. >>a set of points in a plane, each of which is the same distance from a fixed point

equivalent

>conditional statement and its contrapositive are equivalent >converse and inverse are equivalent

right circular cone

A cone with circular base having its apex (highest point) directly above the center of its base

plane

A flat surface; surface which lies evenly with the straight lines itself; may be named by 3 Capital Letters representing points that lie in the plane; or by a letter of the Greek alphabet, such as α alpha (alpha), β beta (beta), or γ gamma (gamma).

bar graph 12.1

A frequency distribution of non-numerical observations

bi-modal 12.2

A histogram with two peaks (modes)

proper subset of a set

A is a proper subset of set if A B and A B; written as A B

probability distribution

A listing, which shows all possible values of a random variable, along with the probabilities that those values will occur, is a probability distribution for that random variable. All possible values are listed, so they make up the entire sample space, and thus the listed probabilities must add up to 1

discrete random variable 12.5

A random variable that can assume only certain fixed values.

empirical probability

A series of repeated experiments provides an empirical probability for an event, which, by "inductive reasoning", is an estimate of the event's theoretical probability. Increasing the number of repetitions increases the reliability of the estimate.

null set

A set with no elements

contraction/ shrink

A size transformation having magnitude k<

dilation/ stretch

A size transformation having magnitude k>1

identity translation

A translation of magnitude 0 leaves every point of the plane unchanged

inverses

A translation of magnitude k, followed by a similar translation of magnitude k but of opposite direction, returns a point to its original position, so these two translations are inverses of each other

Find the unknown side lengths in similar triangles PQR and ABC AC=24 AB=18 BC=30 QR=35 RP=b QP=a

A=21 B=28

empirical rule

About 68% of all data values of a normal curve lie within 1 standard deviation of the mean (in both directions), about 95% within 2 standard deviations, and about 99.7% within 3 standard deviations.

acute triangle

All Angles Acute

equilateral triangle

All Sides Equal

R = {7, 8, . . . , 15, 16}

Although only 4 elements are listed, the ellipsis points indicate that there are other elements in the set. Counting them all, we find that there are 10 elements, so n(R) = 10

acute angle

An angle whose measure is between 0° and 90°

point

An exact location; a dot on a line: Usually represented by a Capital Letter

interior angles on same side of transversal

Angle measures add to 180°

alternate exterior angles

Angle measures are equal.

alternate interior angles

Angle measures are equal.

corresponding angles

Angle measures are equal.

straight angle

Angle that measures 180°

inscribed

Any angle inscribed in a circle has degree measure half of that of its intercepted arc

space figure

Boxes are one kind of

Let T be a translation having magnitude four fifths inch to the right in a direction parallel to the bottom edge of the page. Let r Subscript m be a reflection about line​ m, and let Upper R Subscript p be a rotation about point P having magnitude 40 degrees clockwise. Perform the given transformation on point A of the figure to the right to obtain final image point​ A'.

Choose the figure below that shows the correct transformation of Upper T times Upper R Subscript p.

asymptote 12.5

Close but Never Touching When a curve approaches closer and closer to a line, without ever actually meeting it (as a normal curve approaches the horizontal axis)

"A Prime"

Complement of A, contains all elements that are contained in "U" but are not contained in "A"

binomial probability formula

Define the following quantities. n = the number of repeated trials p = the probability of success on any given trial q = 1−p=the probability of failure on any given trial a = the number of successes that occur

B= {1, 1, 2, 3, 2}

Do not count repeated elements more than once. Set B has only three distinct elements, so n(B) =

triangle size 12ft; 16ft whar is side (c) ft

Find the length of the third side of the right triangle. c=20 ft

antecedent

If

Multiplication Rule of Probability (Event A & B)

If A and B are any two events, then P(A and B)=P(A)⋅P(B|A). If A and B are independent, then P(A and B)=P(A)⋅P(B).

Addition Rule of Probability (for the event "A and B)

If A and B are any two events, then the following holds. P(A or B)=P(A)+P(B)−P(A and B) If A and B are mutually exclusive, then the following holds. P(A or B)=P(A)+P(B)

expected value of x

If a random variable x can have any of the values and the corresponding probabilities of these values occurring are P(x1),P(x2),P(x3),...,P(xn), then E(x)

percentile 12.4

If approximately n percent of the items in a distribution are less than the number x, then x is the nth percentile of the distribution, denoted Pn.

complementary

If the sum of the measures of two acute angles is 90°

side-side-side

If three sides of one triangle are equal, respectively, to three sides of a second triangle, then the triangles are congruent.

angle-side-angle

If two angles and the included side of one triangle are equal, respectively, to two angles and the included side of a second triangle, then the triangles are congruent.

supplementary

If two angles have a sum of 180

side-angle-side

If two sides and the included angle of one triangle are equal, respectively, to two sides and the included angle of a second triangle, then the triangles are congruent.

hypotenuse

In a right triangle, the side opposite the right angle (the longest side)

distinguishable arrangements

In counting arrangements of objects that contain look-alikes, the normal factorial formula must be modified to find the number of truly different arrangements. For example, the number of distinguishable arrangements of the letters of the word DAD is not 3!=6 3 factorial equals 6 but rather 3!2!=3. fraction 3 factorial , over 2 factorial end fraction . equals 3 . The listing below shows how the six total arrangements consist of just three groups of two, where the two in a given group look alike.

Non-Symmetric 12.2

In distributions skewed to the left (a), the data points start low and gradually go up. In distributions skewed to the right (b), data points go up quickly and then gradually go down. In bimodal distribution (c), data points form two peaks.

A lot is in the shape of a triangle. One side is 600ft longer than the shortest​ side, while the third side is 700ft longer than the shortest side. The perimeter of the lot is 4000ft. Find the lengths of the sides of the lot.

Lengths of the sides of the lit are: 900; 1500; 1600 ft.

properties of probability

Let E be an event within the sample space S. That is, E is a subset of S. Then the following properties hold. 1. 0≤P(E)≤1 The probability of an event is a number from 0 through 1, inclusive. 2. P(∅)=0 The probability of an impossible event is 0. 3. P(S)=1 The probability of a certain event is 1.

converting between probability and odds

Let E be an event. • If P(E)=ab, then the odds in favor of E are a to (b−a). • If the odds in favor of E are a to b, then P(E) = a / a+b.

glide reflection

Let rm r sub m be a reflection about line m, and let T be a translation having nonzero magnitude and a direction parallel to m. Then the composition of T and rm Here a reflection followed by a translation is the same as a translation followed by a reflection, so in this case

theoretical probability

Likewise, an established theoretical probability for an event enables us, by "deductive reasoning", to predict the proportion of times the event will occur in a series of repeated experiments. The prediction should be more accurate for larger numbers of repetitions.

Sets of Numbers

Natural numbers (or counting numbers); Whole numbers; Integers; Rational Numbers; Real Numbers; Irrational numbers;

scalene triangle

No Sides Equal

coefficient of variation 12.3

Often this is a more meaningful measure than a straight measure of dispersion, especially when we are comparing distributions whose means are appreciably different.

obtuse triangle

One Obtuse Angle

right triangle

One Right Angle

empirical probability formula

P(E) = #of times event E occurred / # of times the experiment was performed If E is an event that may happen when an experiment is performed, then an empirical probability of event E is given by the following formula

Theoretical Probability Formula

P(E) = P (event) = # of favorable outcomes / total # of outcomes <<>>If all outcomes in a sample space S are equally likely, and E is an event within that sample space

point reflection

Point Q bisects the line segment from a point A to its image A', eh prime comma and for this reason this rotation is sometimes called

regular polygon

Polygons with all sides equal and all angles equal

Label a segment whose endpoints are Upper Q and Upper R. Determine an appropriate name for a segment whose endpoints are Upper Q and Upper R. Choose the correct segment whose endpoints are Upper Q and Upper R Choose an appropriate name for a segment whose endpoints are Upper Q and Upper R.

Q(end-----end)R Q(end-----end)R

Classify the triangle as ​equilateral, isosceles, or scalene. Then classify the triangle as ​right, obtuse, or acute. Triangle with sides 63,49,21

Scalene; Obtuse

K = {3, 9, 27, 81

Set K contains 4 elements, so the cardinal number of set K is 4, and n (K) = 4

(M) = {0}

Set M contains only one element, 0, so n(M) = 1

conditional probability

Sometimes the probability of an event must be computed using the knowledge that some other event has happened (or is happening, or will happen—the timing is not important)

Testing the validity of an Argument with a Truth Table

Step 1 Assign a letter to represent each component statement in the argument. Step 2 Express each premise and the conclusion symbolically. Step 3 Form the symbolic statement of the entire argument by writing the conjunction of all the premises as the antecedent of a conditional statement, and the conclusion of the argument as the consequent. Step 4 Complete the truth table for the conditional statement formed in Step 3. If it is a tautology, then the argument is valid; otherwise, it is invalid.

median 12.2

Step 1 Rank the items (that is, arrange them in numerical order from least to greatest). Step 2 If the number of items is odd, the median is the middle item in the list. Step 3 If the number of items is even, the median is the mean of the two middle items.

set equality (alternative definition)

Suppose A and B are sets: A = B is A B and B A are both true

theorems

The Greeks were the first to insist that all propositions about geometry be given rigorous proofs before being accepted

area

The amount of plane surface covered by a polygon measured in square units

vertex

The common endpoint of the rays of the angle

Conditional Probability Formula

The conditional probability of B given A is calculated as P(B|A) = P(A∩B) / P(A) = P(A and B) / P(A)

Conditional Probability Formula

The conditional probability of B given A is calculated as P(B|A) = P(A∩B) / P(A) = P(A and B) / P(A). If A and B are mutually exclusive, then the following holds. P(A or B)=P(A)+P(B)

disjunction

The disjunction p∨q p or q is false only if both component statements are false.

circumference

The distance around a circle is its circumference (rather than its perimeter)

magnitude

The distance between a point and its image under a translation The measure of angle ABA'

Many mathematical objects that are studied have dimensions that are whole numbers. For​ example, such solids as cubes and icosahedrons have dimension three.​ Squares, triangles, and many other planar figures are​ two-dimensional. Lines are​ one-dimensional, and points have dimension zero. Consider a square with side of length​ one, such as the one shown to the right. The size of a figure is calculated by counting up the number of replicas​ (small pieces) that make it up.​ Here, a replica is the original square with edges of length one. What is the least number of these squares that can be put together edge to edge to form a larger​ square? Square 1

The least number of squares is 4

mean (arithmetic mean) 12.2

The mean of a set of numbers is found by adding all the values in the set and dividing by the number of values 1) most common measure of central tendency "x bar" 2) The mean of n data items x1,x2,..., xn, x sub 1 , comma , x sub 2 , comma dot dot dot comma , x sub n , comma is calculated as follows. "x bar"=(∑x)/n 3) average

Pythagorean triple

The natural numbers 3, 4, and 5 form the Pythagorean triple (3, 4, 5) because they satisfy the equation of the Pythagorean theorem

factorial formula for permutations

The number of permutations, or arrangements, of n distinct things taken r at a time, where r≤n, can be calculated as follows. nPr = n!/(n−r)!

legs

The other two sides, which are perpendicular

center of rotation

The point of intersection of these two nonparallel lines

Conditional probability of B given A

The probability of event B, computed on the assumption that event A has happened, is called the conditional probability of B given A and is denoted P(B|A)

Probability of a Complement (For the Event "NOT E")

The probability that an event E will not occur is equal to 1 minus the probability that it will occur. P(not E)= 1 − P(E) P(E) + P (E') = 1 P(E) = 1 - P(E')

Match the following set with the appropriate description: {1, 3, 5, 7, 9}

The set of odd positive integers less than 10

Match the following set with the appropriate description. {2, 4, 6, 8, 10}

The set of the 5 least positive integer multiples of 2

Symmetric 12.2

The three figures all show symmetric formations. In uniform distribution (a), the data points have the same vertical value. In binomial distribution (b), data points go up, hit a peak, and then go down. In bi-modal distribution (c), data points form two peaks.

random variable

The time spent on homework is an example of a random variable. It is "random" because we cannot predict which of its possible values will occur.

consequent

Then

collinear

Three points that lie on the same straight line

congruent triangles

Triangles that are both the same size and the same shape

isosceles triangle

Two Sides Equal

independent events

Two events A and B are independent events if knowledge about the occurrence of one of them has no effect on the probability of the other one occurring. A and B are independent if: P(B|A)=P(B), or equivalently, P(A|B)=P(A).

mutually exclusive events

Two events A and B are mutually exclusive events if they have no outcomes in common. Mutually exclusive events cannot occur simultaneously.

subsets of a set

U = {1,2,3,4,5}, while A = {1,2,3} Set A is a subset of set B if every element of A is also an element of B

composition or product

We shall use the symbol rm r sub m to represent a reflection about line m, and let us use rn⋅rm r sub n , dot , r sub m to represent a reflection about line m followed by a reflection about line n. We call rn⋅rm

Two angles are supplementary. One is 15 degrees more than four times the other. Find the measures of the angles. Supplementary angles are angles whose sum is 180degrees.

What is the measure of the smaller​ angle? 33 degrees ​(Simplify your​ answer.) What is the measure of the other​ angle? 147 degrees ​(Simplify your​ answer.)

fundamental counting principle

When a task consists of k separate parts and satisfies the uniformity criterion, if the first part can be done in n1 n sub 1 ways, the second part can then be done in n2 n sub 2 ways, and so on through the kth part, which can be done in nk n sub k ways, then the total number of ways to complete the task is given by the following product.

binomial probability formula

When n independent repeated trials occur, where p = probability of success and q = probability of failure with p and q (where q = 1 − p) remaining constant throughout all n trials, the probability of exactly x successes is calculated as follows. P (x) = nCxpxqn−x=n!x!(n−x)! pxqn−x

diameter

a chord that passes through the center

subset

a collection of some of the members; may be all members of the original set, or even none of them, or anywhere in between

conditional statement

a compound statement that uses the connective if...then

range 12.3

a data set is a straightforward measure of dispersion (greatest value in the set) - (least value in the set)

statement

a declarative sentence that is either true or false (not both simultaneously)

probability

a good measure of likelihood

tangent

a line that touches (intersects) the circle in only one point

set

a mathematical term for a group of objects

volume

a measure of capacity of a space figure measured in cubic units

parallelogram

a quadrilateral with two pairs of parallel sides 4-sided figure with both pairs of opposite sides parallel

square

a rectangle with all sides having equal length

chord

a segment whose endpoints both lie on the circle

histogram 12.1

a series of rectangles whose lengths represent the frequencies and placed next to one another

polygon

a simple closed curve made up only of straight line segments.

convex

a simple closed figure that surrounds 2 points A and B inside the figure

normal curve 12.5

a symmetric, bell-shaped curve

tree diagram

a systematic way of listing all the subsets of a given set

stem-and-leaf display 12.1

a tool of exploratory data analysis

central tendency 12.2

a traditional measurement of mode

Simpson's paradox 12.2

a trend appears in different groups of data but disappears when they are combined

right angle

an angle that is 90°

invalid

an argument that is not valid

Find all unknown angle measures in the pair of similar triangles. Note that the triangles are not drawn to scale. a= b=66 c=25 m= n= p=

a=89 b=66 c=25 m=66 n=89 p=25

population 12.1

all items of interest; * includes sample *

truth table

all possible combinations of truth values for the component statements

simple curve

can be drawn without lifting the pencil from the paper and without passing through any point twice

one-part task

can be listed easily; such as tossing a coin

elements

can be numbers, words, objects, etc.

deterministic phenomenon

can be predicted exactly on the basis of obtainable information

standard normal curve

can be used to analyze any normal (or approximately normal) distribution a type of normal curve where the mean is 0 and the standard deviation is 1

random phenomenon

cannot be predicted exactly with obtainable information

descriptive statistics 12.1

collecting, organizing, summarizing, & presenting data (information)

set

collection or group of things; commonly designated using a list within braces, as we have been designating the club

odds

compares the number of favorable outcomes to the number of unfavorable outcomes >>against >>in favor >>>>if all outcomes in a sample space are equally likely, a of them are favorable to the event E, and the remaining b outcomes are unfavorable to E: * odds in favor of E are "a to b"* *odds against E are (b to a)*

tetrahedron

composed of four equilateral triangles, each three of which meet in a point.

octahedron

composed of groups of four regular (i.e., equilateral) triangles meeting at a point.

translation

composite transformation

fallacy of the converse

conditional and converse are NOT equivalent

Decide whether the figure is convex or not convex. Choose the correct answer below.

convex

Natural numbers

counting numbers {1,2,3,4,....}

classes 12.1

data sets

line graph 12.1

demonstrates how a quantity changes >> use a line to connect points

skew lines

do not lie in a common plane, so they are neither parallel nor intersecting

inferential statistics 12.1

drawing inferences or conclusions (making conjectures) about populations on the basis of information from samples

z-score 12.4

each individual item in a sample can be assigned >> the 1st measure of position that we consider >>>If "x" is a data item in a sample with mean "x bar" and standard deviation "s", then the z-score of x is calculated as follows:

size transformation

every point of the image semicircle, such as C', c prime comma was obtained by drawing a line through M and C, and then locating C' c prime such that |MC'|=2|MC|

empirically

experimentally

plane figure

figures that can be drawn completely in the plane of a sheet of paper.

counting

finding the number of objects, of some certain type, that exist

center

fixed point in the middle of a circle

complement of a set

for any set "A" within the universal set "U", the complement of "A", written A', is the set of elements of U that are not elements of "A" A' = {x|x U and x A

Chebyshev's theorem 12.3

for any set of numbers, regardless of hoe they are distributed, the fraction of them that lie within "k" standard deviations of ther mean (where k is greater than 1) is at least ( 1 - 1/k squared)

tree diagram

for tasks that have more than 2 parts is not easy to analyze with a product table

intersecting planes

form a straight line, the one and only line they have in common

dodecahedron

formed by groups of three regular pentagons

fair game

game in which the expected net winnings are zero

circle graph (pie chart) 12.1

graphic alternative to the bar graph

closed curve

has its starting and ending points the same and is drawn without lifting the pencil from the paper

prism

have two faces in parallel planes; these faces are congruent polygons

conditional statement

if I go, then you stay

contrapositive

if the antecedent and the consequent are both interchanged and negated

valid

if the truth of the premises are true forces the conclusion to be true

equivalent statements

if they have the same truth value in every possible situation. The columns of the two truth tables that were the last to be completed will be the same for equivalent statements.

simulation

imitation an important area within probability theory

order

in all cases decide whether order is important, because that determines whether to use permutations or combinations

conditional connective

in no way implies a cause-and-effect relationship

universal set

in set theory the universe of discourse typically designated by the letter U

universe of discourse

includes all things under discussion at a given time

line segment

includes both endpoints and is named by its endpoints

ray

including an initial point with a half-line

existential qualifiers

indicate at least one member: at least one

argument

inductive reasoning we observe patterns to solve problems; > made up of premises and conclusion

raw data

information that has been collected but not yet organized of processed

converse

interchanging the antecedent and the consequent

fallacy of the inverse

invalid argument

transformational geometry

investigates how one geometric figure can be transformed into another

box plot (box-and-whisker plot) 12.4

involves the median (a measure of central tendency), the range (a measure of dispersion), and the first and third quartiles (measures of position), all incorporated into a simple visual display. >> consists of a rectangular box positioned above a numerical scale, extending from Q1 to Q3, with the value of Q2 (the median) indicated within the box, and with "whiskers" (line segments) extending to the left and right from the box out to the minimum and maximum data items

probability of any event

is a number from 0 through 1, inclusive

finite set

is when the cardinal number of a set is a particular whole number (0 or a counting number)

intersecting lines

lie in the same plane and cross each other

parallel line

lie in the same plane and never meet, no matter how far they are extended.

transformation

line m is perpendicular to the line segment AA' eh eh prime and bisects this line segment

listing method

listing the elements, separated by commas, in squiggly brackets Example: {a, e, i, o, u}

pyramid

made of triangular sides and a polygonal base

icosahedron

made up of groups of five regular triangles

compound statement

made up of two or more component statements joined by connectives (not, and, or, if ... then).

theoretically

mathematically

measures of position 12.4

measure an items position within the data set

class mark 12.1

middle value

uniformity criterion

multi-part task; if the number of choices for any particular part is the same no matter which choices were selected for previous parts

negation

must have the opposite truth value from the original

inverse

negating both the antecedent and the consequent the contrapositive of the converse

parallel planes

never meet, no matter how far they extend

skewed to the right 12.2

non-symmetric distribution with a tail extending to the left

negation

of a true statement has the opposite truth value of that statement in all cases: *true is false and false is true*

perimeter

of any polygon is the sum of the measures of the line segments that form its sides. Perimeter is measured in linear units.

truth values of conjunction

of component statements are used to find truth values of compound statements

half-line

one on each side of the point

hexahedron

or cube, is composed of six squares, each three of which meet at a point

frequency distribution 12.1

organized data set that includes many repeated items 1) distinct data value (x) 2) with their frequencies (f)

biconditional

p if and only if q ( p iff q) > is true when both component statements have the same truth value > is false when they have different truth values

geometric construction

page 63

qualitative (non-numerical) data 12.1

page 645

quantitative (numerical) data 12.1

page 645

similar triangles

pairs of triangles that are exactly the same shape but not necessarily the same size.

invariant point

point A and its image, A', eh prime , comma under a certain transformation are the same point

secant

point of tangency. PQ←→ modified p q with left right arrow above which intersects the circle in two points

outcomes

possible results of the experiment

possible sequences

possibly outcomes

line

properties of no thickness and no width & extend indefinitely in 2 directions: may be represented by 2 Capital Letters representing points on that lie or line may also be represented by usually lower case *Cursive l* *subscript is sometimes used*

inclusive disjunction

p∨q means " p or q" or both

ranked data 12.1

quantitative data arranged in numerical order

palindromic

read the same forward and backward

line of symmetry

reflection images about the lines of reflection

n(A); which is read as "n of A"

represents the "cardinal number" of "set A"

symbol ∼

represents the connective not

simulation methods

require huge numbers of random digits, so computers are used to produce them.

radius

segments whose endpoints are in the center and a point on the circle

sample space

set of all possible outcomes the set of all possible outcomes of a probability experiment

Identify the curve as​ simple, closed,​ both, or neither.

simple

frequency polygon classes 12.1

simple plot a single point at the appropriate height for each frequency, connect the point with a series of connected & complete the polygon with segments that trail down to the axis

sample 12.1

some but ordinarily not all, of the items in population

disjunction syllogism

statement is a tautology and the argument is valid

component statements

statements making up a compound statement

euler diagram

technique to check the validity

expected frequency distribution 12.1

the 1st 2 columns of table

rotation

the composition of two reflections about nonparallel lines

Suppose carpet for a 10 ft by 8 ft room costs ​$300. Find the cost to carpet a room 40 ft by 32 ft.

the cost to carpet the room is "$4800"

Venn diagram

the entire region bounded by the rectangle represents the universal set U, and the portion bounded by the circle represents set A

polyhedra

the faces of which are made only of polygons

A search plane carries radar equipment that can detect metal objects​ (like submarine periscopes or plane​ wreckage) on the ocean surface up to 18.5 miles away. If the plane completes a circular flight pattern of 785 miles in​ circumference, how much area will it​ search? (Use 3.14 as an approximation for pi​.) Three concentric circles are shown over the Indian Ocean. A plane lies between the inner and outer most circles on the right side. The circle in between the other two contains a dashed border and denotes the planes flight pattern.

the flight pattern has a total search area of approximately (29,000 square miles)

relative frequency distribution 12.1

the fraction, or % of data set represented by the item 1) If n denotes the total number of items, and a given item, x, occurred f times, then the relative frequency of x is f/n.

upper class limits 12.1

the largest unit in a class of class (10-19) upper class limit is 19

modus tollens

the law of contraposition or indirect reasoning

reasoning by transitivity

the law of hypothetical syllogism

measure of central tendency 12.2

the middle value of the set

symmetry in data sets 12.2

the most useful way to analyze a data set often depends on the distribution

cardinal number (cardinality)

the number of elements in a set: n(A); which is read as "n of A"

number of proper subsets

the number of proper subsets of a set with _____ elements

number of subsets

the number of subsets of a set with _____ elements

(n) factorial

the quantity of n formula is defined as: n! = n * (n - 1) * (n - 2) * .... * 2 * 1

sides

the rays of an angle

reflection preserves collinearity

the reflection image of a line is also a line the reflection image of a line is also a line. We express this by saying that reflection preserves collinearity.

observed (empirical) frequencies 12.1

the results from the 1st 2 columns

Match the following set with the appropriate description. {..., -4, -3, -2, -1}

the set of all negative integers

lower class limits 12.1

the smallest possible data values within the respective classes of class (10-19) lower class limit is 10

dispersion (spread) of data 12.3

the spread of information

tautology

the statement is always true no matter what the truth value of the components

cumulative frequency 12.2

the sum of the frequencies for that class and all previous classes

quartile 12.4

the three values ( Q1,Q2, q sub 1 , comma . q sub 2 , comma and Q3 q sub 3 ) that divide a data set into four (approximately) equal-sized parts >For any set of data (ranked in order from least to greatest): 1) The second quartile, Q2, is just the median, the middle item when the number of items is odd, or the mean of the two middle items when the number of items is even. 2) The first quartile, Q1 is the median of all items below Q2. 3) The third quartile, Q3 is the median of all items above Q2.

angle

the union of two rays that have a common endpoint

Venn diagrams

the universal set is represented by a rectangle, and other sets of interest within the universal set are depicted by circular regions

mode 12.2

the value that occurs most often

weighted mean 12.2

the weighted mean of a group of (weighted) items is the sum of all products of items times weighting factors, divided by the sum of all weighting factors

sphere

three-dimensional analogue of a circle

conjunction p^q

to be true, both p and q must be true

protractor

tool used to measure angles

surface area

total area that would be covered if the space figure were "peeled" and the peel laid flat

isometry

transformations in which the image of a figure has the same size and shape as the original figure