Finite Math

7.3 Elementary Event

An event with only one element. Ex. E={2}

Union

One set AND another subset combined without repetitions.Symbol: U. Ex. N: {1,2,3} P: {3, 4, 5} N U P= {1,2,3,4,5}

Converse

The antecedent and consequent reverse positions. Ex. The converse of P-->Q is Q-->P

Morgan's Law I

The negation of the conjunction (and) is the disjunction (or). Ex. ~(P/\Q)=P\/Q Basically, /\+ ~= \/

The Empty Set

The set with no objects in it. It is a subset of any other set, even of itself. The symbol is a slashed zero, like this (/)

Argument

A group of premises and conclusion to be determined valid or invalid. Takes the symbolic form of a fraction with the premises placed as the numerator and the conclusion the denominator. Ex. P, P\/Q/ Q

Set

A well-defined collection of objects in which it is possible to determine if a given object belongs to the collection. Ex. 4 belongs to {3,4,5) 6 does not belong to {3, 4, 5} The symbol for 'belongs to' is like a c with a center prong. To negate that, just run a slash through the symbol.

Associative Law

Ex. (P\/Q)\/R= P\/(Q\/R) If all the connectives are the same, their position doesn't matter.

When is an "or" statement true?

If at least one component of the statement is true. Ex. P\/Q Read as "P or Q" or "P in disjunction with Q." P | Q | P\/Q| T T T T F T F T T F F F

Invalid Argument

If even one premise is false, the conclusion is invalid.

Inverse

Negate both the antecedent and consequent, which remain where they are. Ex. The inverse of P-->Q is ~P-->~Q

9.1 Frequency Polygon

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9.1 Histogram

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Tree Diagram

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Combination Formula

C(n,r)= n!/(n-r)!r! also written as n=S and r=E Ex. C(5,2)= 5!/(5-2)!2! = 5!/3!2! = 5*4*3*2*1/3*2*1 * 2*1 = 5*4/2*1 = 20/2 =10

7.3 Sample Space

The set of all possible outcomes for an experiment. Ex.Possible Outcomes= 1,2,3,4 Sample space= {*1,2,3,4*}

Negation

When a statement is negated (symbol: ~) Ex. ~P: I am NOT here.

Contradiction

A compound statement that is always false. Ex. P/\~P Read as P and not P, which is impossible.

Tautology

A compound statement that is always true. Ex.P\/~P Read as P or not P, which means any option is true.

7.4 Union Rule

P(E U F)= p(E)+P(F)-P(E/\F)

7.3 Basic Principle of Probability

Also known as the relative frequency interpretation of probabilities, it applies when all outcomes in S are equally probable. Equation: P(E)= n(E)/ n(S) Read as: the probability of event E equals the number of events divided by the number of sets.

7.3 Event

An event (symbolically: E) is a subset of S, a sample space (set of all possible outcomes). Ex. S= {1,2,3,4,5,6} E= {2}

Combination to Permutation Formula

C(n,r)= P(n,r)/r!

Morgan's Law II

The disjunction is equivalent to the conjunction of the negation. Ex. P\/Q= ~(P/\Q)

Compound Statement

Two or more statements joined together with a connective, each represented by a letter. Ex. P /\ Q: I am here and I am walking around.

Set Builder Notation

Used to describe a set of objects that share a common property. Ex. {3,4,5}= {x|x is an integer >2/<6)} The set containing three, four and five an be described with a set builder notation that states any number in the given set is an integer greater than two and less than 6.

Venn Diagram

Visual used to explore the relationships of sets to one another composed of overlapping circles.

When is an "and" statement true?

When every component of the statement is true. Ex. P /\Q Read as "P and Q" or "P in conjunction with Q." P | Q | P/\Q T T T T F F F T F F F T

Conditional Statement

When the connective "If/then" is used. Ex. P--->Q. Read as "If P, then Q." or "P implies Q." A conditional statement is only false when the antecedent (in this case P) is true but the consequent (Q) is false. The negation of a conditional is a conjunction: P--> Q= ~P\/Q

Connective

Words used to connect two or more statements to create a compound statements. Connectives commonly used in finite math are "and" (symbol: /\) (AKA a conjunction), "or" (symbol: \/)(AKA a disjunction), and "If/Then"(symbol: -->)(AKA an implication).

Subset

A given sets could also belong to a larger set. Every set is a subset of another set.The symbol for "is a subset of" is an underlined C. To negate this, slash it. Ex. {Germany, France} is a subset of {Europe} or {3, 4, 5} is a subset of {1,2,3,4,5}

Statements

A phrase that can be answered simply with a yes or no. Represented by a letter. Ex. P: I am walking around.

Universal Set

A set containing every number you are concerned with for specific problem without repetitions. Symbol: U.

Intersection

A set that shows the objects that both sets have in common. Symbol: upside down U. Ex. N: {1,2,3} P: {3,4,5} The intersection of N and P is= {3}

9.1 Frequency Table

A table listing the intervals of information and also a tally of how many times each interval is represented in the given data.

Valid Argument

All premises must be true (tautology) to make the conclusion valid.Determined through a truth table, simplified in the example Ex. ~P, P\/Q/ Q P | Q | ~P/\(P\/Q)--> Q T T T* T F T* F T T* F F T* Column containing the truth values for the complete statement is completely true, thus this is a VALID ARGUMENT.

7.4 Complement

Any number, within the Universal Set, that is outside of the given set. Symbolized through a comma, the complement of set A is A'. U: {1, 2, 3, 4, 5} N: {1, 2, 3} N': {4, 5}

Biconditional Statement

Conjunction of conditional converse (basically saying "if and only if"). True if both elements are true or both are false. Ex. P<-->Q Read as "P if and only if Q".

Binomial Probability

Equation: P( x successes in n trials)=C(n,x)* p raised to the x* (1-p) raised to the n-x

Distributive Law

Ex. P/\ (Q\/R)= (P/\Q) \/ (P/\R) The conjunction of a disjunction is the conjunction of two disjunctions.

Commutative Law

If the statements in a conjunction or disjunction are flipped around but the connective remains the same, than it still means the same thing. Ex, P/\Q=Q/\P Ex. P\/Q=Q\/P

Multiplicative Principle

N number of choices must be made with M1 ways to make choice 1, M2 ways of making choice 2, etc.

Probability Formula

P(E)= N(E)/N(S)

7.4 Complement Rule

P(E)=1- P(E')

Permutation Formula

P(n,r)= n!/(n-r)! also written as n=S and r=E Ex. P(5,2)= 5!/(5-2)! =5!/3! = 5*4*3*2*1/3*2*1 = 5*4 = 20

Permutation to Combination Formula

P(n,r)=C(n,r)*r!

7.3 Empirical Probablity

Probability based on surveys or historical data.

Truth Table

Table used to determine whether a statement is true by looking at the truth of each of its components in any given case. Ex. Statement: Q--->P Truth Table: Q| P | Q--->P| T T T T F F F T T F F T

How do you find the number of subsets to a set?

Take the number of objects of in a set and make that number the degree over 2. Ex. {3,4,5} 3 objects. Thus, 2 to the third= 8 Subsets.

9.1 Mean

The Average. Sum of x's divided by the number of x's.

Contrapositive

The converse of the inverse. Reverse the positions of the antecedent and consequent, then negate them. Ex. The contrapositive of P-->Q is ~Q-->~P