CORE 131 Mathematics I Study Guide

Inductive Reasoning

Inductive Reasoning

Drawing a general conclusion from repeated observations of specific examples.

Conjecture

Another name for a general conclusion.

Deductive Reasoning

Applying general principles to specific examples.

Is this an example of Inductive or Deductive Reasoning?

Our house is made of brick. My neighbor's house is made of brick. Therefore, I can conclude that all of the houses in the neighborhood are made of brick.

Inductive. Goes from specific examples to a general statement based off said examples.

Is this an example of Inductive or Deductive Reasoning?

All keyboards have the @ symbol. I have a keyboard. So I will assume my keyboard also has the @ symbol.

Deductive. Goes from a general example and applies it to a specific situation.

Number Sequence

A list of numbers having a first number, a second number, a third number, and so on.

Arithmetic Sequence

A sequence in which each term is found by adding/subtracting the same number to the previous term.

Common Difference

The difference between the terms in an arithmetic sequence. Found by addition or subtraction.

Geometric Sequence

A sequence in which each term is found by multiplying or dividing the previous term by the same number.

Common Ratio

The constant ratio of any term and the previous term in a geometric sequence. Found by multiplying/dividing.

Is this sequence Arithmetic, Geometric, or Neither?

1, 5, 9, 13, 17, 21

Arithmetic

Is this sequence Arithmetic, Geometric, or Neither?

3, 6, 12, 24, 48, 96

Geometric

Is this sequence Arithmetic, Geometric, or Neither?

7,776, 1,296, 216, 36, 6, 1

Geometric

Is this sequence Arithmetic, Geometric, or Neither?

54, 45, 36, 27, 18, 9

Arithmetic

Is this sequence Arithmetic, Geometric, or Neither?

95, 84, 74, 60, 57, 41

Neither

Is this sequence Arithmetic, Geometric, or Neither?

3, 8, 24, 66, 108, 136

Neither

What is the common difference/ratio of this sequence?

3, 6, 12, 24, 48, 96

4 - Difference

What is the common difference/ratio of this sequence?

6,561, 729, 81, 9

9 - Ratio

Successive Differences

The process to determine the next term of a sequence using subtraction to find a common difference.

Set

A collection of objects called elements/numbers.

Example: A = {2, 4, 6, 8}

What are the three ways to designate a set?

1. Word Description → (The set containing ...)

2. Listing Method → {2, 4, 6, 8}

3. Set-Builder Notation → (x|x is ...)

What does this symbol mean?

∈

Element of

What does this symbol mean?

∉

Not an element of

Finite Set

A set where elements CAN be counted

Infinite Set

A set where elements CANNOT be counted

Cardinal Number/Cardinality

The number of elements in a set.

Denoted by n(A)

What is the cardinality of this set?

A = {9, 10, 11, 12, 15, 18, 20}

n(A) = 7

What is the cardinality of this set?

Z = {10, 20, 30}

n(Z) = 3

Write the answer to this set using set-builder notation.

- The set of counting numbers between 5 and 10.

x|x is a counting number between 5 and 10.

Write the answer to this set using the listing method.

- The set of counting numbers between 5 and 10.

{6, 7, 8, 9}

True or False?

10∈ {10, 11, 12, 13, 14, 15}

True

True or False?

3∈ {1, 2, 4, 5, 10}

False

True or False?

16∉ {15, 20, 25, 30, 35}

True

Empty Set/Null Set

Set with NO elements. Written as {} or ∅.

Equal Sets

Sets with the exact same elements

If sets are equal, they are also equivalent!

Equivalent Set

Sets with the same number of elements

Are these sets equal, equivalent, neither, or both?

A = {9, 10, 16, 20}

B = {9, 10, 16, 20}

Both.

- They have the exact same elements and the same number of elements

- n(A) = 4 & n(B) = 4

Are these sets equal, equivalent, neither, or both?

Q = {1, 3, 5, 7, 9}

P = {2, 4, 6, 8, 10}

Equivalent

- They have the same number of elements, but they do not have the exact same elements.

- n(Q) = 5 & n(P) = 5

Are these sets equal, equivalent, neither,or both?

M = {5, 10, 15, 20}

N = {11, 12, 13, 21, 22, 23}

Neither

- They have neither the same elements nor the same number of elements

- n(M) = 4 & n(N) = 6

Are these sets equal, equivalent, neither, or both?

B = {9, 18, 27, 36}

C = {27, 9, 38, 18}

Both

- They both have the exact same elements. Even though they are not in the same order, they are still equal & equivalent sets.

- n(B) = 4 & n(C) = 4

Natural/Counting Numbers

{1, 2, 3, 4, 5, 6, 7 , 8, 9}

Whole Numbers

{0, 1, 2, 3, 4, 5, 6, 7, 8, 9}

Integers

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

Rational Numbers

- {3/5, -7/9, 0, 5, -101/1, 3.6,}

- Rational numbers also include decimals that terminate or repeat.

- Natural Numbers, Whole Numbers, and Integers are all rational numbers

Irrational Numbers

- {√2, π, 5.9284.....}

Real Numbers

Natural Numbers, Whole Numbers, Integers, Rational Numbers, and Irrational Numbers

Subsets

For Sets A and B, Set A is a Subset of Set B if every element in Set A is also in Set B. It is written as A ⊆ B.

Proper Subsets (⊂)

For Sets A and B, Set A is a Proper Subset of Set B if every element in Set A is also in Set B, but Set A does not equal Set B. (𝑨 ≠ 𝑩) It is written as A ⊂ B

- If something is a proper subset, it will also always be a subset.

Is set Q a subset, proper subset, both, or neither of set R?

Q = {1, 3, 6, 9}

R = {1, 3, 6, 9, 12}

Both - all of the elements in Q are in set R, but set Q does not equal set R. So it is both a subset, and a proper subset.

Is set D a subset, proper subset, both, or neither of set E?

D = {100, 200, 300, 400}

E = {500, 600, 700, 800}

Neither. Set D does not have any of the same elements that set E does.

Is set X a subset, proper subset, both, or neither of set Y?

X = {5, 10, 15, 20}

Y = {20, 10, 15, 5}

Subset. Set X has all the same elements that set Y has.

Let A = {q, r s}

Let B = {p, r, s, t}

Let C = {r, s}

True or False: B ⊆ A

False

Let A = {q, r s}

Let B = {p, r, s, t}

Let C = {r, s}

True or False: C ⊆ A

True

Let A = {q, r s}

Let B = {p, r, s, t}

Let C = {r, s}

True or False: C ⊂ B

True

Let A = {q, r s}

Let B = {p, r, s, t}

Let C = {r, s}

True or False: B ⊂ C

False

Let A = {q, r s}

Let B = {p, r, s, t}

Let C = {r, s}

True or False: ∅ ⊆ B

True - The empty set is a subset AND proper set of every set.

⊆

Symbol for Subset

⊂

Symbol for proper subset

∩

Symbol for intersection of sets

∪

Symbol for union of sets

Intersection of Two Sets

The set of elements that are common to both of the sets

Union of Two Sets

A set that brings together all the elements of both sets together

What is the union of set A and set B?

A = {l, m, n, o, p}

B = {m, p, q, s, t}

A∪B = {l, m, n, o, p, q, s, t}

What is the intersection of set A and set B?

A = {l, m, n, o, p}

B = {m, p, q, s, t}

A∩B = {m, p}

Cartesian Product of Sets

- A set of pairs (a, b) of elements from two sets A and B.

- Multiple the number of elements in Set A, by the number of elements in Set B, to get your product.

What is the cartesian product of Set A X Set B?

A = {1, 5, 9}

B = {2, 6}

AxB = {(1,2), (1,6), (5,2), (5,6), (9,2), (9,6)}

n(AxB) = 6

What is the cartesian product of Set B X Set A?

A = {1, 5, 9}

B = {2, 6}

BxA = {(2,1), (2,5), (2,9), (6,1), (6,5), (6,9)}

n(BxA) = 6

Complement of a set

The set of all elements in the universal set that are not in a given set.

What is the Complement of Set A?

U = {a, b, c, d, e, f, g}

A = {a, d, g}

A' = {b, c, e, f}

Subtracting Sets

The set A−B consists of elements that are in A but not in B

What is A - B?

A = {1, 3, 5, 9, 11, 13, 15}

B = {1, 3, 6, 9, 12, 14, 15}

n(A-B) = {5, 11, 13}

What is B - A?

A = {1, 3, 5, 9, 11, 13, 15}

B = {1, 3, 6, 9, 12, 14, 15}

n(B-A) = {6, 12, 14}

Number of Subsets in a Set

The number of subsets in a set can be found by: 2ⁿ

(n being the number of elements in a set)

Number of Proper Subsets in a Set

The number of subsets in a set can be found by: 2ⁿ-1

(n being the number of elements in a set)

How many subsets does Set S have?

S = {34, 65, 96}

8

How many proper subsets does Set S have?

S = {101, 202, 303, 404, 505, 606,}

63

What is the Complement of Set Y?

U = {x, r, q, t, o}

Y = {o}

Y' = {x, r, q, t}

Statement

A true or false declarative sentence. NOT questions.

Compound Statements

Statements made using logical connectors. (and, but, or, if, not, then)

Is this sentence a statement, a compound statement, or neither?

My brother works at Pearson and Hardman.

Statement

Is this sentence a statement, a compound statement, or neither?

What time are you leaving today?

Neither

Is this sentence a statement, a compound statement, or neither?

After the movie, we can go get ice cream, or we can go get sno-cones.

Compound statement

^

Means "and"

Also called a conjunction

V

Means "or"

Also called a disjunction

~

Means "not"

Also called a negation

Write a conjunction sentence of these statements using symbols.

a = The Midterm is this week.

b = It will be sunny tomorrow.

a^b

Write a disjunction sentence of these statements using symbols.

a = The Midterm is this week.

b = It will be sunny tomorrow.

aVb

How would you write "The midterm is this week and it will not be sunny tomorrow" using symbols?

a = The Midterm is this week.

b = It will be sunny tomorrow.

a^~b

How would you write "Neither is the Midterm this week nor will it be sunny tomorrow" using symbols?

a = The Midterm is this week.

b = It will be sunny tomorrow.

~(aVb)