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Finite Mathematics

Inductive Reasoning Observation of Specific Examples
Deductive Reasoning Proving a specific conclusion from one or more general statements
Rounding If digit to the right is 5 or more, round the digit up 1 and change everything to the right to zeros. If less than five, don't change and make everything to the right zeros
Roster Method Listing items of a set between curly braces {}
Set Builder Notation B={x|x is all of the days of the week that begin with T}
The Empty Set {} or 0 with a line through it.
ϵ Is an Element of
ϵ with a / through it Is NOT an element of
Natural Numbers Counting Numbers starting with 1 and going to infinity
Cardinality of a set n(A) The number of distinct elements of a set. Said as n of A
Equivalence of a set Two (or more) sets have the same cardinal number
Finite Set If it is the empty set or if its cardinal number is a natural number
Set Equality If both sets have the same elements, regardless of order or repetition, they are equal.
A sideways U above half an equals sign B means A is a subset of B
A sideways U above half an equals sign, with a / through it means A is NOT a subset of B
Sideways U (no line) Proper Subset A is a proper subset of B if it is a subset of B and is NOT equal to B
Number of Subsets with n elements 2^n
Number of Proper Subsets (2^n)-1
Statement Can be true or false, but not both simultaneously
All A are B There are no A that are not B
Some A are B There exists at least one A that is B
No A are B All A are not B
Some A are not B Not all A are B
All A are B - negated Some A are not B
Some A are B - negated All A are not B
OR v
But Yet Nevertheless ALL AND
Antecedent Comes before the connective -->
Consequent Comes after the connective -->
If p then q p-->q
q if p p-->q
p is sufficient for q p-->q
q is necessary for p p-->q
p only if q p-->q
only if q, p p-->q
<--> If and only if
p if and only if q p<-->q
q if and only if p p<-->q
If p then q and if q then p p<-->q
p is necessary and sufficient for q p<-->q
q is necessary and sufficient for p p<-->q
parenthesis, bi-conditional, conditional, conjunctive, negation order of operations for logic
commas indicate grouping in written logic She is happy and wealthy, only if she is married. (h^w)<-->m
Conjunction Truth Only true when both simple statements are true.
Disjunction Falsity Only false when both simple statements are false
Conditional Falsity Only false when the antecedent is true and the consequence is false
Biconditional Truth Only true when both sides are the same.
Factor = Divisor If the remainder is 0, then x is divisible by y, y is a factor of x and y is a divisor of x
Prime Number >1 and divisible only by itself and 1
Composite Number >1 and divisible by itself, 1 and at least 1 other number
Fundamental Theorem of Arithmetic Every composite number can be expressed as the product of prime numbers in 1 and only 1 way.
Relatively prime Greatest Common Divisor is 1
Find Greatest common divisor 1. Factor both numbers completely. 2. Find the factors common to both and use the smallest exponent 3. Multiply those numbers
Find Least Common Multiple 1. Factor the numbers 2. Use EVERY factor, with the largest exponent only 3. Multiply these numbers
Whole Numbers 0 and natural numbers
Integers 0 and natural numbers and negatives of natural numbers
Rational Numbers All numbers that can be expressed as a fraction where both numerator and denominator are integers and the denominator is not 0
Repeating Decimal as a fraction 1. n= repeating decimal 2. Multiply it by 10 for one repeat, 100 for two, etc 3. Subtract equation 1 from equation 2 4. Solve for N
a/b * c/d (a*c)/(b*d)
(a/b)/(c/d) (a*d)/(b*d)
Irrational Numbers Numbers whose decimals are neither terminating or repeating. sqrt 2 and Pi
simplifying square roots sqrt of 18 = sqrt 9 * sqrt 2 therefore sqrt 18 =3sqrt2 and division works the same way.
Real Numbers Set of Rational Numbers and Irrational Numbers
Exponent Rules To multiply, add To divide subtract Raising to a power, multiply
Closure Every possible result of the operation exists in the set of numbers included in the operation
Commutative Property Doesn't matter what order you do it in.
Associative Property (A#B)#C = A#(B#C)
Identity Property Operation on any object in the set with the Identity Element results in the original element
Inverse Property When operated on with the particular operation, it results in the identity element for that set.
Identity Element of Addition 0
Identity Element of Multiplication 1
Identity Element of Division 1
Identity Element of Subtraction 0
Group Closed (under the operation given) Is Associative under the operation Has an identity element each element has an inverse in the set
congruent in a modular system a (3 equals) b (mod m) a and b give same remainder when divided by m
Modular addition Modulo M system add a + b. If it's less than M, that's the answer If it's greater than M, the answer is the remainder when it is divided by M
Equivalent Graphs Have the same number of vertices connected in the same way
Degree of vertex Number of edges connected to the vertex. A loop counts as two.
Euler Path Travels through every edge once and only once
Euler Circuit Travels through every edge once and only once and begins and ends on the same vertex
Rules of Euler Graphs 1. If it has exactly two odd vertices, it has 1 path that starts at one odd vertex and ends at the other. 2. If it has all even vertices, it has a least one euler circuit and it can begin on any vertex 3. If more than two odd vx, no path or circuit
Hamilton Path Passes through each vertex of a graph once and only once
Hamilton Circuit Passes through each vertex and begins and ends at the same vertex
Number of Hamilton-Circuits (n-1)!
Created by: ahuddle