Section 1 1 Statements and Conditional Statements Statement a declarative sentence that is either true of false but not both also called a proposition to establish truth write mathematical proof to establish false provide counterexample Techniques of Exploration guesswork and conjectures Examples Constructing appropriate examples is very important can only say appears to be true if you can t find a counter Use of prior knowledge cooperation and brainstorming Conditional Statements given the form If P then Q where P and Q are sentences For this conditional statement P is called the hypothesis and Q is called the conclusion Think of If P then Q to be false if I lied and to be true If I did not lie Rational numbers Q are those real numbers that can be written as a quotient of two integers Irrational numbers are those real numbers that cannot be written as a quotient of two integers Natural numbers N consist of the positive whole numbers Integers Z consist of zero the positive whole numbers and the negatives of the positive whole numbers each integer is a rational and a real number integers are closed under addition multiplication and subtraction Section 1 2 Constructing Direct Proofs Definition an agreement that a particular word or phrase will stand for some object property or other concept that we expect to refer to often Even integer an integer a is even provided that there exists an integer n such that a 2n Odd integer an integer a is odd provided that there exists an integer n such that a 2n 1 Know Show Table steps 1 Identify the hypothesis P first step and the conclusion Q goal of the conditional statement 2 Start with things we know such as definitions 3 ask backward and forward questions 4 use algebra substitution whatever to finish proving Mathematical Proof a convincing argument that a certain mathematical statement is necessarily true Guidelines for Mathematical proof 1 Begin with a carefully worded statement of the theorem or result to be proven a State theorem b Skip a line and write Proof in italics or bold 2 Begin the proof with a statement of your assumptions a we assume that 3 Use the pronoun we 4 Use italics for variables when using word processor 5 Display important equations and mathematical expressions 6 Tell the reader when the proof has been completed a this completes the proof or end of proof symbol Section 2 1 Statements and Logical Operators a Algebra or mathematical expressions should be separated by a space and centered Logical operator on mathematical statements is a word or combination of words that combines one or more mathematical statements to make a new mathematical statement Compound statement a statement that contains one or more operators Conjunction P and Q or P Q Only true when both P and Q are true Disjunction P or Q or PvQ Only true when at least one of Por Q is true False only when both are false Negation not P or P negation of P is true only when P is false and vice versa implication If P then Q or P Q Only false when P is true and Q is false Other ways to expresses P Q If P then Q Q if P P implies Q Whenever P is true Q is true P only if Q Q is true Whenever P is true Q is necessary for P means that if P is true then Q is necessarily true is equivalent to P is sufficient for Q means if you want Q to be true it is sufficient to show P is true A reason for a conclusion but not the only reason know how to construct truth table Biconditional statement P if and only if Q or P Q or Q P P Q can be expressed as P if and only if Q P is necessary and sufficient for Q and P implies Q and Q implies P Tautology a compound statement S that is true for all possible combinations of truth values of the component statements that are part of S Contradiction a compound statement that is false for all possible combinations of truth values of truth values of the component statements that are part of S Section 2 2 Logically Equivalent Statements Logically equivalent two statements are this provided that they have the same truth values for all possible combinations of truth values for all variables appearing in the two expressions In this case we write X Y and say that X and Y are logically equivalent Converse of conditional statement P Q is the conditional statement Q P Contrapositive of the conditional statement P Q is the conditional statement Q P Section 2 3 Open Sentences and Sets Set a well defined collection of objects that can be thought of as a single entity of itself typically listed between braces listing the elements of a set inside braces is called the roster method of specifying the elements Variable a symbol representing an unspecified object that can be chosen from a given set U the universal set for the variable Constant specific member of the universal set Symbols in the subset of The set A is a subset of a set B provided that each element of A is an element of B In this case we write A B and also say that A is contained in B Open sentence or predicate or a propositional function a sentence P x1 x2 Xn invoving variables x1 x2 xn with the property that when specific values from the universal set are assigned to x1 x2 xn then the resulting sentence is either true or false That is the resulting sentence is a statement the truth set solution set of an open sentence with one variable is the collection of objects in the universal set that can be subsituted for the variable to make the predicate a true statement Set Builder notation the set is defined by stating a rule that all elements of the set must satisfy Ex If P x is a predicate in the variable x then the notation is x U P x whiwhc stands for the set of all elements x in the universal set U for which P x is true means such that Empty set when a set contains no elements is designated by the symbol Section 2 4 Quantifiers and Negations Quantifiers there exists V upside down A For all or for every Universal quantifier Existential quantifier Negation of Quantified Statements Random definitions Perfect Square the natural number n is a perfect square provided that there exists a natural number k such that n k2 Multiple of three an integer n is a multiple of 3 provided that there exists an integer k such that n 3k Statements with more than one quantifier look at examples on pg 72 Negations of multiple quantifiers Pythagorean triples