Math

= the set of all rational or irrational numbers

Real Numbers

: A - B = {x : x ∈ A and x ∉ B}

Relative Complement

: The lesson introduces the important topic of sets

Sets and Subsets

are common in plants and in some animals, notably molluscs.

Spirals

The _____ is the statement "Not A or Not B".

negation of "A and B"

The _____ is the statement "Not A and Not B."

negation of "A or B"

the ______ becomes "A and not B".

negation of "if A, then B"

is denoted by {∅}.

null set

set that contains no elements is called a

null set or an empty set

means engaging in a task for which the solution method is not known in advance

Problem Solving

is the logic of compound statements built from simpler statements using so-called Boolean connectives

Propositional Logic

operate on propositions or truth values instead of on ,numbers

Propositional or Boolean operators

c. 570-c. 495 BC) explained patterns in nature like the harmonies of music as arising from number, which he took to be the basic constituent of existence.

Pythagoras

: A common ratio between two subsequent terms.

Ratio(r)

={ the set of all terminating or repeating decimals}

Rational Numbers

: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100...

Square numbers

: A △ B = x ∈ A and x ∉ B or x ∈ B and x ∉ A= (A−B)∪(B−A)

Symmetric Difference

used to refer to an object that is invariant under some transformations; including translation, reflection, rotation or scaling.

Symmetry

: Each number in a sequence, where the first number is called 1st term, the second number is the 2nd term, the nth number is the nth term

Term(n)

are patterns formed by repeating tiles all over a flat surface.

Tessellations

: 1, 3, 6, 10, 15, 21, 28, 36, 45, 55...

Triangular numbers

take 1 operand (e.g., −3);

Unary operators

: A ∪ B = {x : x ∈ A or x ∈ B}

Union

If two sets are given, a set can be formed by using all the elements of the two sets.

Union of Sets:

is a graphical tool in which we use overlapping circles to visually presentation among some given sets information

Venn diagram

= {0, 1, 2, 3, 4, 5, . . .}

Whole Numbers

The _____ states that p is true if and only if (IFF) q is true.

biconditional p ↔ q

The _______combines two propositions to form their logical conjunction.

binary conjunction operator "∧" (AND)

The _____combines two propositions to form their logical disjunction.

binary disjunction operator "∨" (OR)

The _____ combines two propositions to form their logical "exclusive or" (exjunction?)

binary exclusive-or operator "⊕" (XOR)

take 2 operands (eg 3 × 4)

binary operators

A set is usually denoted by a

capital letter

Its _____ : ¬q → ¬ p

contrapositive

Its _____ is: q → p.

converse

Two sets are _______ if they have exactly the same elements in them.

equal

golden ratio is also called the

golden mean or golden section

, is a mathematical concept that people have known about since the time of the ancient Greeks.

golden ratio

, which he defined as plausible reasoning in solving mathematical problems. He encouraged the problem solver to guess the result and then use strict logical reasoning to prove his guess.

heuristics

The _____ states that p implies q

implication p → q

The _______ of two sets is the set of elements that are common to both sets.

intersection

Its _____ is: ¬p → ¬q.

inverse

tries to find a pattern in a sequence by taking successive differences until a pattern is found and then working backwards.

method of successive difference

combines one or more operand expressions into a larger expression

operator or connective

A is a situation in which a person is seeking somegoal and for which a suitable course of action is not immediately apparent

problem

is simply a statement (i.e., a declarative sentence) with a definite meaning, having a truth value that's either true (T) or false (F) (never both, neither, or somewhere in between).

proposition (p, q, r, ...)

Fibonacci investigated (in the year 1202) was about how fast _______ could breed in ideal circumstances.

rabbits

Plants often have ________, as do many flowers and some groups of animals such as sea anemones

radial or rotational symmetry

Is a well-defined collection of objects

set

element of a set is usually denoted by a

small letter

If every element in Set A is also in Set B, then Set A is a _____ of Set B.

subset

n - a1 - d -

term, first term, common difference

n - a1 - r -

term, first term, common ratio

The ______transforms a prop. into its logical negation.

unary negation operator "¬" (NOT)

The _____ of two sets is the set of elements that belong to one or both of the two sets.

union

Common value in arithmetic sequence

Common difference

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

Integers

developed two laws of negation

AUGUSTUS DE MORGAN

: A U (A ∩ B) = A and A ∩ (A U B) = A

Absorption laws

: A sequence where a constant number is added from the previous term to get to the next term. ( 2, 4, 6, 8, ...) (2, 2+2=4 , 4+2=6, 6+2=8)

Arithmetic Sequence

(A U B) U C= A U (B U C) and (A ∩ B) ∩ C= A ∩ (B ∩ C)

Associative Laws:

Animals mainly have _______ as do the leaves of plants and some flowers such as orchids.

Bilateral symmetry or mirror symmetry

A U B = B U A and A ∩ B = B ∩ A

Commutative law

: A' = {x: x ∈ U and x ∉ A}

Complement

: (Ac)c =A

Complement Laws

: Let U be a universal set and A be any subset of U, then the elements of U which are not in A

Complement of Set

The ________ of set B in set A is the complement of B in A. If A and B are any two sets, then the _______ of B in A is the set of all elements in A which are not in B. It is denoted by A - B

Difference of Two Sets (Relative Complement):

: A common difference between two subsequent terms.

Difference(d)

Two sets A and B are said to be disjoint, if their intersection is a null set

Disjoint Sets:

: A U (B ∩ C) = (A U B) ∩ (A U C) and A ∩ (B U C) = (A ∩ B) U (A ∩ C)

Distributive Laws

: A U U= U and A = ∅ ∅

Domination laws

Each object in a set is called an _____ of the set.

Element

(c. 494-c. 434 BC) to an extent anticipated Darwin's evolutionary explanation for the structures of organisms.

Empedocles

presented a thought experiment on the growth of an idealized rabbit population

Fibonacci

______ is anumber that can be represented by a regular geometrical arrangement of equally spaced points. If the arrangement forms a regular polygon, the number is called a _______

Figurate number ,polygonal number

is found in the echinoderms, the group that includes starfish, sea urchins, and sea lilies.

Fivefold symmetry

"There exists x such that A(x)"

For every x, not A(x)

developed set theory, his theory was not initially accepted because it was radically different

GEORGE CANTOR

: A sequence where a constant number is multiplied to the previous term to get to the next term. ( 2, 4, 8, 16, ...) ( 2 , 2x2=4, 4x2=8, 8x2=16)

Geometric Sequence

developed an algebra of logic

George Boole

1.6180339887

Golden number

1, 7, 18, 34, 55, 81, 112, 148, 189, 235

Heptagonal numbers:

: 1, 6, 15, 28, 45, 66, 91, 120, 153, 190

Hexagonal numbers

: AUA = A and A U A = A

Impotent laws

: A ∩ B = {x : x ∈ A and x ∈ B}

Intersection

: A set can be formed by using all the common elements of two given sets.

Intersection of Sets

= the set of all nonterminating, nonrepeating decimals

Irrational Numbers

devised a simple way to diagram set operations (Venn Diagrams)

John Venn

introduced the Fibonacci sequence to the western world with his book

Leonardo Fibonacci

Book of Leonardo Fibonacci

Liber Abaci

are patterns that show relationship between figures or images

Logic patterns

is a tool for working with complicated compound statements

Mathematical Logic

={1, 2, 3, 4, 5, . . .}

Natural Numbers or Counting Numbers

Early Greek philosophers attempted to explain order in ______, anticipating modern concepts.

Nature

The opposite of a given mathematical statement

Negotiation

: 1, 9, 24, 46, 75, 111, 154, 204, 261, 325...

Nonagonal numbers

is a pattern or sequence in a series of numbers. This pattern generally establishes a common relationship between all numbers.

Number pattern

: 1, 8, 21, 40, 65, 96, 133, 176, 225, 280...

Octagonal numbers

start with "1" at the top, then continue placing numbers below it in a triangular pattern. Each number is the numbers directly above it added together

Pascal's triangle

is also present in language in terms of morphological rules in plurals of nouns, verb tenses, as well as in metrical rules of poetry.

Pattern

include symmetries, trees, spirals, meanders, waves, foams, tessellations, cracks and stripes

Patterns in nature

: 1, 5, 12, 22, 35, 51, 70, 92, 117, 145

Pentagonal numbers

(c. 427-c. 347 BC) argued for the existence of natural universals.

Plato

"For all x, A(x)"

There exist x such that not A(x)

Explicit Formula for Geometric Sequence,

an = a1 ∙ r ^ (n-1)

Explicit Formula for Arithmetic Sequence

an = a1 + d (n - 1)