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# GAC1

### Finite Mathematics

Question | Answer |
---|---|

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 |

AND | ^ |

OR | v |

NOT | ~ |

But Yet Nevertheless | ALL AND |

IF-THEN | --> |

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