Math 113 _ Chapters 1 & 2

Fibonacci sequence

1,1,2,3,5,8,13,21,34,55,89,144,... NOT Arithmetic because there is no fixed difference. First two terms of the Fibonacci sequences and 1, 1 and each subsequent is the sum of the previous two.

Well-defined set

Must be able to identify whether or not an object is an element of that set

Statement

a sentence that is either true or false, but not both

Recursive pattern

after one or more consecutive terms are given, each successive term of the sequence is obtained from the previous term(s)

Biconditional

"p if, and only if, q"

Equal Sets

two sets are equal if, and only if, they contain exactly the same elements

Law of Denying the Consequent (modus Tollens)

With a conditional accepted as true but having a false conclusion, the hypothesis must be false.

Universal Quantifiers

Words: all, every, & no Refer to each and every element

Figurative Number

examples of sequences that are neither arithmetic nor geometric

Conjectures

statements or conclusions that have not been proven

Set Intersection

the intersection of two sets A and B, is all elements common to both A and B

Inductive reasoning

the method of making generalizations based on observations and patterns

Guess and Check

First guess at an answer using as reasonable a guess as possible. Then we check to see whether the guess is correct. If not, the next step is to learn as much as possible about the answer based on the guess before making a next guess.

Cartesian Product

For any sets A and B, the Cartesian product of A and B, written AxB, is the set of all ordered pairs such that the first component of each pair is an element of A and the second component of each pair is an element of B

Fundamental Counting Principle

If event M can occur in m ways and, after it has occurred, event N can occur in n ways, then event M followed by event N can occur in mn ways

One-to-one Correspondence

If the elements of sets p and s can be paired so that for each element of p there is exactly one element of s and for each element of s there is exactly one element of p, then the two sets p and s

Law of Detachment (Modus Ponens)

If the statement "if p, then q" is true and p is true, then q must be true

Equivalent Sets

Two sets A and B are equivalent, written A~B if and only if there exists a one-to one correspondence between the sets -have the same number of elements and are not necessarily equal but they could be

Set Union

Union of two sets A and B, Written A U B, is the set of all elements in A or in B

Existential Quantifiers

Words: some & there exists at least one Refer to one or more, or possibly all, of the elements

Counterexample

an example that shows a conjecture is false

Problem Solving

finding a way out of difficulty, a way around an obstacle, attaining an aim which was not immediately attainable

Logically Equivalent

if, and only if, two statements have the same truth values in every possible situation

Cardinal Number

n(s) number of elements in set

Deductive reasoning

the use of definitions, undefined terms, mathematical axioms that are assumed to be true, and previously proved theorems, together with logic to prove these conjectures

4-Step problem solving process

1. Understanding the problem 2. Devising a plan 3. Carrying out the plan 4. Looking back

Truth Tables

Show all possible true-false patterns for statements

Chain rule (Transitivity)

If "if p, then q" and "if q, then r" are true, then "if p, then r" is true

Quantifiers

Are more complicated to negate Words: all, some, every, & there exists

Geometric sequence

2,4,8,16,32 Each successive term of a geometric sequence is obtained from its predecessor by multiplying by a fixed nonzero number, the ratio.

Subset

B is a subset of A, if and only if, every element of B is an element of A

Relative Complement

Complement of A relative to B, written B - A, is the set of all elements in B that are not in A

Arithmetic sequence

Sequence in which each successive term from the second term on is obtained from the previous term by the addition (+) or subtraction (-) of a fixed number

Conditionals or Implications

Statements expressed in the form "if p, the q" denoted by p -> q. Can also be read "p implies q"

Set Complement

The complement of set F, written F with a line on top, is the set of all elements in the universal set U that are not in F