# discrete mathematical structures

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Discrete Mathematical Structures. 离散数学结构. Bernard Kolman Robert C. Busby Sharon Cutler Ross. 《 离散数学 》 教学组. Chapter 2 Logic. 2.1 Propositions and Logical Operations 2.2 Conditional Statements 2.3 Methods of Proof 2.4 Mathematical Induction. 推理的符号化（形式化） 推理规则举例： 假言推理（ modus ponens ） - PowerPoint PPT Presentation

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• Discrete MathematicalStructuresBernard KolmanRobert C. BusbySharon Cutler Ross

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2.1 Propositions and Logical Operations2.2 Conditional Statements2.3 Methods of Proof2.4 Mathematical InductionChapter 2 Logic

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modus ponens 2011 2011

p q p ------------ q

p => q p ------------ q

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Syllogism 2011 2011

x(A(x) -> B(x)) A(a) ------------ B(a)

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Reasoning in MathematicsGottfried Wilhelm Leibnizs (July 1, 1646 November 14, 1716) Symbolic thought: The only way to rectify our reasonings is to make them as tangible as those of the Mathematicians, so that we can find our error at a glance, and when there are disputes among persons, we can simply say: Let us calculate [calculemus], without further ado, to see who is right. It is obvious that if we could find characters or signs suited for expressing all our thoughts as clearly and as exactly as arithmetic expresses numbers or geometry expresses lines, we could do in all matters insofar as they are subject to reasoning all that we can do in arithmetic and geometry. For all investigations which depend on reasoning would be carried out by transposing these characters and by a species of calculus.Formal Logic George Boole (2 November 1815 -8 December 1864 ) invented Boolean Logic.

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2.1 Propositions and Logical Operations1) Statement or Proposition A statement or proposition () is a declarative sentence that is either true or false, but not both.Ex. Which of the following are statements?(a) The earth is round.(b) 2 + 3 = 5.(c) Do you speak English?(d) 3 x = 5(e) Take two aspirins.(f) The temperature on the surface of the planet Venus () is 800F.(g) The sun will come out tomorrow.YesYesNo, it is a question.No, it is a declaration sentence, not a statement.No, it is a command.Yes, it is a statement.Yes, it is a statement.

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Proposition examples (compound propositions):2 is an even number and 3 is an odd number.2 is an even number and 3 is an even number.2 is an odd number and 3 is an odd number.2 is an odd number and 3 is an even number.2 is an odd number and I am a student.Pattern of the proposition: with connective and Symbol: /\ (conjunction)Truth of the pattern, or the truth table of /\

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Compound proposition examples:2 is an even number or 3 is an odd number.2 is an even number or 3 is an even number.2 is an odd number or 3 is an odd number.2 is an odd number or 3 is an even number.2 is an odd number or 2 is an even number.

Pattern of the proposition: with connective or Symbol: \/ (disjunction)Truth of the pattern, or the truth table of \/

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Compound proposition examples:If I am hungry, then Ill eat.If it is raining, then 2+3 = 5.If it is raining, then 2+3 = 4.If 2 is an even number, then 3+3 = 4.If 2 is an odd number, then 3 +3 = 4.If 2 is an odd number, then the sun rise on the west.Pattern of the proposition: with connective if then Symbol: => (implication, antecedent, consequent)Truth of the pattern, or the truth table of =>

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Compound proposition examples2 is not an odd number.2 is not an even number.Pattern of the proposition: with connective not Symbol: ~Truth of the pattern, or the truth table of ~

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Equivalence of propositions p => q and ~p \/ qBidirectional p < = > qTautologiesAbsurdity Videos:Inventing on Principlehttp://worrydream.com/ABriefRantOnTheFutureOfInteractionDesign/

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2.1 Propositions and Logical Operations2) Proposition variable and Compound statementsIn Logic, the letters p, q, r, denote proposition variable (), they are replaced by statements.p: The sun is shining today.q: It is cold.Statement or proposition variables can be combined by logical connectives () to obtain compound statements().Ex. andp and q: The sun is shining today and it is cold.

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2.1 Propositions and Logical Operations3) Negation of statement If p is a statement, the negation () of p is the statement not p, denote by ~p.~p is the statement it is not the case that p.If p is true, ~p is false. If p is false, ~p is true.The truth table() is a table that shows the truth valuesof a compound statement in terms of its component parts.FTTF~ppThe truth table

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2.1 Propositions and Logical Operations3) Negation of statementnot is not a connective, it does not join two statements.not p is not compound statement.not is a unary operation for collection of statements.~p is statement if p is.Ex. Give the negation of the following statements: (1) p: 2 + 3 > 1.(2) q: it is cold.Solution:(1) ~p: 2 + 3 1.(2) ~q: it is not cold.

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2.1 Propositions and Logical Operations4) Conjunction of statements If p and q are statements, the conjunction of p and q is p and q, denote by pq. stands for connective and.The compound pq is true if both p and q are true, otherwise, it is false.The truth table of pqTFFFT TT FF TF Fpqp q

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2.1 Propositions and Logical Operations4) Conjunction of statements Ex. From the conjunction of p and q for each of the following.(a) p: It is snowing, q: I am cold.(b) p: 2 < 3, q: -5 > -8.(c) p: It is snowing, q: 3 < 5.Solution:(a) pq: it is snowing and I am cold.(b) pq: 2 < 3 and -5 > -8.(c) pq: It is snowing and 3 < 5.

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2.1 Propositions and Logical Operations5) Disjunction of statements If p and q are statements, the disjunction of p and q is p or q, denote by pq. denotes the connective or.The compound pq is true if at least one of p or q is true, it is false when both p and q are false.The truth table of pqTTTFT TT FF TF Fpqp q

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2.1 Propositions and Logical Operations5) Disjunction of statements Ex. From the disjunction of p and q for each of the following.(a) p: 2 is a positive integer q: 21/2 is a rational number.(b) p: 2 + 3 5 q: London is the capital of France.Solution:(a) pq: 2 is a positive integer or 21/2 is a rational number. p is true, q is false, so pq is true.(b) pq: 2 + 3 5 or London is the capitcal of France. p and q are false, so pq is false.

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2.1 Propositions and Logical Operations6) Compound statement The compound statement has many component parts, both of component parts are connected by connective.pq(pr) is compound statement and has three compound parts (propositions): p, q and r.pq(pr) = pqrTTTTTFFFTTFFTFFFTTTTTFTFT T TT T FT F TT F FF T TF T FF F TF F Fpq(pr)q(pr)prpqr

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2.1 Propositions and Logical Operations7) The truth table If a compound statement s has n component statements, there will need to be 2n rows in the truth table for s. its truth table may be systematically constructed in the following way.(1). The first n columns of the table are labeled by statements, further columns are included for all intermediate expression, the last column is for the full statement.(2). Under each of the first n headings, we list the 2n possible n-tuples of truth values for the n component statements.(3). For each of the remaining columns, we compute the remaining truth values in sequence.

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2.2 Conditional StatementsIf p and q are statements, the compound statement if p then q denoted by p q, is called a conditional statement (), or implication ().To p q, p is called the antecedent () or hypothesis (), and q is called the consequent () or conclusion().Ex. Write the implication p q for each of the following.(a) p: I am hungry.q: I will eat.(b) p: It is snowing.q: 3 + 5 = 8Solution:(a) If I am hungry, then I will eat.(b) If it is snowing, then 3 + 5 = 8.

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2.2 Conditional Statementsp q: p implies q; q if p; p only if q.p is a sufficient condition for q, q is necessary condition for p.If p q is a implication, the converse () of p q is the implication q p, and the contrapositive () of p q is the implication ~q ~p.Ex. Give the converse and contrapositive of the implication If it is raining, then I get wet.Solution:p: It is raining; q: I get wet.the converse is q p: If I get wet, then It is raining.the contrapositive is ~q ~p: If I dont get wet, then it is not raining.

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2.2 Conditional StatementsIf p and q are statements, the compound statement p if and only if q, denoted by p q, is called an equivalence () or biconditional ().The connective if and only if is denoted by the symbol .conditional statement (), or implication ().The truth table of p q is following.The truth table of p qTFFTT TT FF TF Fp qp qp q is true only when both p and q are true or when both p and q are false.The equivalence p q can also be stated as p is a necessary and sufficient condition for q.

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2.2 Conditional StatementsEx. Compute the truth table of the statement (p q) (~q ~p).Solution:TTTTTFTTFFTTFTFTTFTTT TT FF TF F(p q) (~q ~p)~q ~p~p~qp qp q

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2.2 Conditional StatementsA statement that is true for all possible values of its propositional variables is called a tautology(/).A statement that is always false is called a contradiction (/) or an absurdity ().A statement that can be either true or false, depending on the truth values of its propositional variables, is called a contingency ().Ex. (a) (p q) (~q ~p) is a tautology.(b) p ~p is an absurdity.(c) (p q) (p q) is a contingency.

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2.2 Conditional StatementsIf p q is a tautology, p and q are logically equivalent (), or simply equivalent ().p is equivalent to q is denoted by p q.Ex. p q q p.The truth table for (p q) (q p) shows the statement is a tautology.TTTTTTT FTTTFT TT FF TF F(p q) (q p)q pp qp q

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2.2 Conditional StatementsEx. p q is equivalent to (~p q).Column 1 and 3 in the above table show that for any truth values of p and q, p q and (~p q) have the same truth values.TFTTFFTTTFTTT TT FF TF F

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