Contemporary ∗†

∈ and /∈The symbol ∈ is used to indicate that an object is an element of a set. The symbol ∈ is used toreplace the words ”is an element of.”

The symbol /∈ is used to indicate that an object is not an element of a set. The symbol /∈ is used toreplace the words ”is not an element of.”

Definition of a Set’s Cardinal Number

The cardinal number of a set A, represented by n(A), is the number of distinct elements in set A.The symbol n(A) is read ”n of A.”

Definition of Equivalent Sets

Set A is equivalent to set B means that set A and set B contain the same number of elements. Forequivalent sets, n(A) = n(B)

One-To-One correspondences and equivalent sets

1. If set A and set B can be place in a one-to-one correspondences, then A is equivalent toB : n(A) = n(B).

2. If set A and set B cannot be place in a one-to-one correspondences, then A is not equivalent toB : n(A) 6= n(B).

Equality of Sets

Set A is equal to set B means that set A and set B contains exactly the same elements, regardless ofthe order or possible repetition of elements. We symbolize the equality of sets A and B using thestatement A = B.

∗Prepared by James Buzzard†Pearson,Central Texas College Custom Edition of Thinking Mathematically,Blitzer,2011

A Subset of a Set

Set A is a subset of set B, expressed asA ⊆ B,

if every element in set A is also an element in set B.

A Proper Subset of a Set

Set A is a proper subset of set B, expressed as A ⊂ B, if set A is a subset of set B ans sets A and Bare not equal (A 6= B)

The Empty Set as a Subset

1. For any set B, ∅ ⊆ B.

2. For any set B other than the empty set, ∅ ⊂ B.

Number of Subsets

The number of subsets of a set with n elements is 2n.

Number of Proper Subsets

The number of proper subsets of a set with n elements is 2n − 1

Complement of a Set

The complement of set A, symbolized by A′, is the set of all elements in the universal set that are notin A. This idea can be expressed in set-builder notation as follows:

A′ = {x|x ∈ U and x /∈ A}.

Intersection of Sets

The intersection of sets A and B, written A ∩B, is the set of elements common to both set A and setB. This definition can be expressed in set-builder notation as follows:

A ∩B = {x|x ∈ A and x ∈ B}.

Union of Sets

The union of sets A and B, written A ∪B, is the set of elements that are members of set A or of setB or of both sets. This definition can be expressed in set-builder notation as follows:

A ∪B = {x|x ∈ A or x ∈ B}.

The Empty Set in the Intersection and Union

For any set A,

1. A ∩∅ = ∅

2. A ∪∅ = A.

Formula for the cardinal number of the union of two finite sets

n(A ∪B) = n(A) + n(B)− n(A ∩B)

Read as:The number of elements in A or B is the number of elements in A plus the number of elements in Bminus the number of elements in A and B.

De Morgan’s Laws

(A ∩B)′ = A′ ∪B′ : The complement of the intersection of two sets is the union of thecomplements of those sets.

(A ∪B)′ = A′ ∩B′ : The complement of the union of two sets is the intersection ofthe complements of those sets.

Statement of Symbolic LogicName Symbolic Form Common English TranslationNegation ∼ p Not p. It is not true that p.Conjunction p ∧ q p and q. p but q.Disjunction p ∨ q p or q.Conditional p→ q If p, then q. p is sufficient for q. q is necessary for p.Biconditional p↔ q p if and only if q. p is necessary and sufficient for q.

Dominace of Connectives

1. Negation, ∼, (least dominant)

2. Conjunction, ∧, and Disjunction, ∨ (same level of dominance)

3. Conditional, →

4. Biconditional, ↔ (most dominant)

Definition of Conjunction

Conjunctionp q p ∧ qT T TT F FF T FF F F

A conjunction is true only when both simple statements are true.

Definition of Disjunction

Disjunctionp q p ∨ qT T TT F TF T TF F F

A disjunction is false only when both component statements are false.

Definition of the Conditional

Conditionalp q p→ qT T TT F FF T TF F T

A conditional is false only when the antecedent is true and the consequent is false.

Definition of the Biconditional

Biconditionalp q p↔ qT T TT F FF T FF F T

A biconditional is true only when the component statements have the same truth value.

Variations of the Conditional statement

Name Symbolic Form English TranslationConditional p→ q If p, then q.Converse q → p If q, then p.Inverse ∼ p→∼ q If not p, then not q.Contrapositive ∼ q →∼ p If not q, then not p.

Negation of a Conditional Statement

The negation of p→ q is p ∧ ∼ q. this can be expressed as

∼ (p→ q) ≡ p ∧ ∼ q

De Morgan’s Law’s and Negations

1. ∼ (p ∧ q) ≡∼ p ∨ ∼ qThe negation of p ∧ q is ∼ p ∨ ∼ q. To negate a conjunction, negate each component statementand change and to or.

2. ∼ (p ∨ q) ≡∼ p ∧ ∼ qThe negation of p ∨ q is ∼ p ∧ ∼ q. To negate a disjunction, negate each component statementand change or to and.

Euler Diagrams and Arguments

1. Make an Euler diagram for the first premise.

2. Make an Euler diagram for the second premise on top of the one for the first premise.

3. The argument is valid if and only if every possible diagram illustrates the conclusion of theargument. If there is even one possible diagram that contradicts the conclusion, this indicatesthat the conclusion is not true in every case, so the argument is invalid.

Divisibility

If a and b are natural numbers, a is divisible by b if the operation of dividing a by b leaves aremainder of 0. This is the same as saying b is a divisor of a, or b divides a. All three statements aresymbolized by writing

b|a.

Finding the Greatest Common Divisor

To find the greatest common divisor of two or more numbers,

1. Write the prime factorization of each number.

2. Select each prime factor with the smallest exponent that is common to each of the primefactorizations.

3. Form the product of the numbers from step 2. The greatest common divisor is the product ofthese factors.

Finding the Least Common Multiple

To find the least common multiple of two or more numbers,

1. Write the prime factorization of each number.

2. Select each prime factor that occurs, raised to the greatest power to which it occurs, in thesefactorizations.

3. Form the product of the numbers from step 2. The least common multiple is the product ofthese factors.

Expressing a Repeating Decimal as a Quotient of Integers

1. Let n equal the repeating decimal.

2. Multiply both sides of the equation in step 1 by 10 if one digit repeats, by 100 if two digitsrepeats, by 1000 if three digits repeat, and so on.

3. Subtract the equation in step 1 from the equation in step 2.

4. Divide bith side of the equation in step 3 by the number in front of n and solve for n.

Exponent Rules

Product Rule bm · bn = bm+n

Power Rule (bm)n = bm·n

Quotient Rule bm

bn= bm−n

Zero Rule b0 = 1

Negative Exponent Rule b−m = 1bm

Scientific Notation

A positive number is written in scientific notation when it is expressed in the form

a× 10n,

where a is a number greater than or equal to 1 and less than 10 (1 ≤ a < 10) and n is an integer.

General Term of an Arithmetic Sequence

The nth term (the general term) of an arithmetic sequence with the first term a1 and the commondifference d is

an = a1 + d(n− 1).

Definition of a Geometric Sequence

A geometric sequence is a sequence in which each term after the first is obtained by multiplying thepreceding term by a fixed nonzero constant. The amount by which we multiply each time is calledthe common ratio of the sequence.

General Term of a Geometric Sequence

The nth term (the general term) of a geometric sequence with the first term a1 and the common ratior is

an = a1rn−1

Common Formula for Percent Application

A = P ·B, this reads as: A is P percent of B

Calculating Income Tax

1. Determine your adjusted gross income:

adjusted gross income = gross income - adjustments.

2. Determine your taxable income:

taxable income = adjusted gross income - (exemptions + deductions).

3. Determine the income tax:

Income tax = tax computation - tax credits.

Finding Percent Increase or Percent Decrease

1. Find the fraction for the percent increase or the percent decrease

amount of increase

orignal amountor

amount of decrease

orignal amount

2. Find the percent increase or the percent decrease by expanding the fraction in step 1 as apercent.

Calculating Simple Interest

Interest = principal× rate× time

I = Prt

The rate, r, is expressed as a decimal when calculating simple interest.

Calculating Future Value for Simple Interest

The future value, A, of P dollars at simple interest rate r (as a decimal) for t years is given by

A = P (1 + rt)

Formulas for Compounding Interest

After t years, the balance, A,in an account with principal P and annual interest rate r (in decimalform) is given by the following formulas:

1. For n compounding periods per year: A = P(1 + r

n)nt

2. For continuous compounding: A = Pert

Calculating Present Value

If A dollars are to be accumulated in t years in an account that pays rate r compounded n times peryear, then the present value, P , that needs to be invested now is given by

P =A(

1 +r

n

)nt .

Effective Annual Yield

Suppose that an investment has a nominal interest rate, r, in decimal form, and pays compoundinterest n mes per year. The Investment’s effective annual yield, Y , in decimal form is given by

Y =(

1 +r

n

)n− 1

The decimal form of Y given by the formula should then be converted to a percent.

Value of an Annuity: Interest Compounded

Once a YearIf P is the deposit made at the end of each year for annuity that pays an annual interest rate r (indecimal form) compounded once a year, the value, A, of the annuity after t years is

A =P [(1 + r)t − 1]

r.

n times per yearIf P is the deposit made at the end of each year for annuity that pays an annual interest rate r (indecimal form) compounded n times per year, the value, A, of the annuity after t years is

A =

P

[(1 +

r

n

)nt− 1

]( rn) .

Regular Payments Needed to Achieve a Financial Goal

The deposit, P , that must be made at the end of each compounding period into an annuity that paysan annual interest rate r (in decimal form) compounded n times per year in order to achieve a valueof A dollars after t years is

P =A( rn

)[(

1 + rn)nt − 1

]

Loan Payment Formula for Fixed Installment Loans

The regular payment amount, PMT , required to replay a loan of P dollars paid n times per year overt years at an annual rate r is given by

PMT =P( rn

)[1−

(1 + r

n)−nt]

Angles of a Polygon

The sum of the measures of the angles of a polygon with n sides is

(n− 2)180◦

Formulas for Geometry

Object Area Volume Perimeter Surface Area

Rectangle A = lw V = lwh P = 2w + 2l SA = 2lw + 2lh+ 2wh

Square A = s2 V = s3 P = 4s SA = 6s2

Parallelogram A = bh P =4∑

k=1

side k

Triangle A = 12bh P =

3∑k=1

side k

Trapezoid A = 12h(a+ b) P =

4∑k=1

side k

Circle A = πr2 C = πd or C = 2πr

Right Circular Cylinder V = πr2h SA = 2πr2 + 2πrh

Cone V = 13πr2h

Sphere V = 43πr3

Trigonometric Ratios

sin θ =side opposite angle θ

hypotenuse

cos θ =side adjacent angle θ

hypotenuse

tan θ =side opposite angle θside adjacent angle θ

Rules of Traversability

1. A graph with all even vertices is traversable. One can start at any vertex and any where onebegan.

2. A graph with two vertices is traversable. One must start at either of the odd vertices and finishat the other.

3. A graph with more than two odd vertices is not traversable.

The Fundamental Counting Principle

The number of ways in which a series of successive things can occur is found by multiplying thenumber of ways in which each thing can occur.

Factorial Notation

If n is a positive integer, the notation n! is the product of all positive integers from n down though 1.

n! = n(n− 1)(n− 2) · · · (3)(2)(1)

0! = 1 by definition

Combinations of n things taken r at a time

The number of possible combinations if r items are taken from n items is

nCr =n!

(n− r)!r!.

Permutations of n things taken r at a time

The number of possible permutations if r items are taken from n items is

nPr =n!

(n− r)!.

Permutations of DuplicateThe number of permutations of n items, where p items are identical, q items are identical, r items areidentical, and so on, is given by

n!

p!q!r!....

Computing Theoretical Probability

If an event E has n(E) equally likely outcomes and its sample space S has n(S) equally likelyoutcomes, the theoretical probability of event E, denoted by P (E), is

P (E) =number of outcomes in event E

total number of possible outcomes=n(E)

n(S)

Computing Empirical Probability

The empirical probability of event E is

P (E) =observed number of times E occurs

total number of observed occurrences.

Complement Rules of Probability

• The probability that an event E will not occur is equal to 1 minus the probability that it willoccur

P (not )E = 1− P (E)

• The probability that an event E will occur is equal to 1 minus the probability that it will notoccur.

P (E) = 1− P (not E

Using set notation, if E ′ is the complement of E, then P (E ′) = 1−P (E) and P (E) = 1−P (E ′).

Or Probabilities with Mutually Exclusive Events

If A and B are mutually exclusive events, then

P (A or B) = P (A) + P (B).

Using set notation, P (A ∪B) = P (A) + P (B).

Or Probabilities with Events that are Not Mutually Exclusive

If A and B are not mutually exclusive events, then

P (A or B) = P (A) + P (B)− P (A and B).

Using set notation,P (A ∪B) = P (A) + P (B)− P (A ∩B).

Probability to Odds

If P (E) is the probability of an event E occurring, then

1. The odds in favor of E are found by taking the probability that E will occur and dividing bythe probability that E will not occur.

Odds in favor of E =P (E)

P (not E)

2. The odds against E are found by taking the probability that E will not occur and dividing bythe probability that E will occur.

Odds against E =P (not E)

P (E)

The odds against E can also be found by reversing the ratio representing the odds in favor of E.

Odds to Probability

If the odds in favor of event E are a to b, then the probability of the event is given by

P (E) =a

a+ b.

And Probabilities with Independent Events

If A and B are independent events, then

P (A and B) = P (A) · P (B)

The Probability of an Event Happening at Least Once

P (event happening at least once) = 1− P (event does not happen)

And Probabilities with Dependent Events

If A and B are dependent events, then

P (A and B) = P (A) · P (B given that A has occurred).

A Formula for Conditional Probability

P (B|A) =n(B ∩ A)

n(A)=

number of outcomes common to B and A

number of outcomes in A

The Mean

The mean is the sum of the data items divided by the number of items.

Mean =

∑x

n,

where∑x represents the sum of all the data items and n represents the number of items.

Calculating the Mean for a Frequency Distribution

Mean = x̄ =

∑xf

n,

wherex represents a data valuef represents the frequency of the data value.∑xf represents the sum of all the products obtained by multiplying each data value by its frequency.

n represents the total frequency of the distribution.

Position of the Median

If n data items are arranged in order, from smallest to largest, the median is the value in the

n+ 1

2

position.

The Midrange

The midrange is found by adding the lowest and highest data values and dividing the sum by 2.

Midrange =lowest data value + highest data value

2

The Range

The range, the difference between the highest and lowest data values in a data set, indicates the totalspread of the data.

Range = highest data value - lowest data value

Computing the Standard Deviation for a Data Set

1. Find the mean of the data items.

2. Find the deviation of each data item from the mean:

data item - mean.

3. Square each item:(data item - mean)2.

4. Sum the squared deviations: ∑(data item - mean)2.

5. Divide the sum in step 4 by n− 1, where n represents the number of data items:∑(data item - mean)2

n− 1.

6. Take the square root of the quotient in step 5. This value is the standard deviation for the dataset.

Standard deviation =

√∑(data item - mean)2

n− 1

Computing z-Score

A z -score describes how many standard deviations a data item in a normal distribution lies above orbelow the mean. The z -score can be obtained using

z -score =data item - mean

standard deviation

Data item above the mean have a positive z -scores. Data items below the mean have negativez -scores. The z -score for the mean is 0.

Percentiles

If n% of the item in a distribution are less than a particular data item, we say that the data item isin the nth percentile of the distribution.

Margin of Error in a Survey

If a statistic is obtained from a random sample of size n, there is a 95% probability that it lies within1√n× 100% of the true population percent, where± 1√

n× 100% is called the margin of error.

Finding the Percentage of Data Items between Two Given Items in a Normal Dis-tribution

1. Convert each given data item to a z -score:

z =data item - mean

standard deviation.

2. Use 12.14 to find the percentile corresponding to each z -score in step 1.

3. Subtract the lesser percentile from the greater percentile and attach a % sign.

Computing the Correlation Coefficient by Hand

The following formula is used to calculate the correlation coefficient, r:

r =n(∑xy)− (

∑x)(∑y)√

n(∑x2)− (

∑x)2√n(∑y2)− (

∑y)2

.

In the formula,

n = the number of data points, (x, y)∑x = the sum of the x-values∑y = the sum of the y-values∑xy = the sum of the product of x and y in each pair∑x2 = the sum of the squares of the x-values∑y2 = the sum of the squares of the y-values(∑

x)2

= the square of the sum of the x-values(∑y)2

= the square of the sum of the y-values

Writing the Equation of the Regression line by Hand

The equation of the regression line isy = mx+ b

where

m =n(∑xy)− (

∑x)(∑y)

n(∑x2)− (

∑x)2

and b =

∑y −m(

∑x)

n