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