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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
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