《Discrete Mathematic with Applications》读书笔记一

Chapter1 The logic of compound statement

1.1 Logic Form and Logic Equivalence.

Translate the natural english argument to logic notation.

such as if p or q, then r.

Def1:

Proposition(statement):  A sentence that is true of false but not both.

exp he is a student. (is not a proposition, the truth and falsity depends on he)

x+y>0 (is not a proposition, the truth and falsity depends on variable X and Y)

Compound statement(proposition)

~ not (~p  negation of p)

^ and(p^q   conjunction of p and q)

V or (p V q  disjunction of p an q)

Order of operation: ~  > ^  = V

English word, and ,but(^)  , not (~),  or (V) 

Neither p nor q   (~p and ~q)

for inequality a < x < b.

The point of specifying x, a and b to particular real numbers is to ensure that sentence such as "x <a" and "x >=b"

are either true or false and hence that they a statements(proposition).

Truth Value

The negation of a statement is a statement that exactly expresses what it would mean for the statement to be false.

So the negation of a statement has opposite truth value from the statement.

intuitive logic, or sometimes mean exclusive(p or q but not both)   sometimes inclusive (p or q or both)

In formal logic, the or means inclusive

xor means exclusive

Exp 1.1.5 Construct the truth table for Exclusive or (XOR)

(p V q) ^ ~(p ^ q)

Exp 1.1.6 Truth table for (p ^ q) V ~r

Logical Equivalence

Statement forms(the same truth values for each possible substitution of statements)

Statements(logically equivalent forms when identical component statement variables are used to replace identical component statements.)

Morgan's Law

Neither ... nor...  =   ~p and ~q

De Morgan's Law are frequently used in writing computer program.

In English sentence could be written more compactly Such as

p= Jim is tall and thin

~= Jim is not tall and thin.

In this case, it doesn't violate the Morgan's Law, Because in formal Logic the 'and', 'or' must connect the statement(proposition). It can be used to connect with the sentence fragment.

When you use the Morgan's Law, apply it with the full statement between 'and', 'or'.

Tautologies(重言式，恒真命题） an Contradiction（悖论）

According to the definition. The truth of the tautology and the falsity of the contradiction is depends on the logical structure of the statements themselves.

It does nothing to do with the meaning of the statement.

Exp1.1.13

  

1.2 Conditional Statements

if p ,then q

p hypothesis

q conclusion

conditional statements

-> is a connective  used for join statements to create new statements.

the only combination of circumstances in which you would call a conditional sentence false occurs when the hypothesis is true and the conclusion is false

p is true, and q is false.

if the hypnosis is false,

the condition statement is true called vacuously true or true by default.

the order of operations (->) is lower than (～，^, V)

p implies q (suggest that q can be logically deduced from p, and it often not the case). Be carefully!

Representation of If-Then As Or

p -> q = ~p V q

The Negation of the Conditional Statement

~(p -> q) = p ^ ~q

The Contrapositive of a Conditional Statement

The contrapositive statement sometimes used for logical proof or logic deduction.

The Converse and the Inverse of a Conditional Statement

Only If and the Biconditional

p iff q  =  (p -> q) ^ (q -> p)

Since p -> q = ~p V q

p <-> q = (p -> q) ^ (q -> p)

= (~p V q) ^ (~q V p)

So all the logic of ->  , <-> can be presented by ~ ^ V

Necessary(必要） and Sufficient（充分） Conditions

r is a necessary and sufficient condition for s also means (if s ,then r) and (if r, then s) =>  r if and only if s

Remarks

1 In logic, a hypothesis and conclusion are not required to have related subject matters.

2 In informal language, simple conditionals are often used to mean biconditionals.

1.3 Valid and Invalid Arguments

In mathematics and logic an argument is not a dispute. It is a sequence of statements ending in a conclusion.

Determine whether an argument is valid - whether the conclusion follows necessarily from the preceding statements.(Depending on the form, not on its content)

  

A valid argument is that the truth of its conclusion follows necessarily or inescapably or by logical form alone from the truth of its premises.

Modus Ponens and Modus Tollens

An argument form consisting of two premises and a conclusion is called a syllogism.

first and second premises are called the major and minor premises.

Modus ponens.(推理式）

It is a valid argument.

Modus tollens(proof by contradiction 否定式)

Additional Valid Argument Forms: Rules of Inference

A rule of inference is a form of argument that is valid.

Modus ponens and modus pollens are both rules of inference.

Generalization

Specialization

Elimination

Transitivity

Proof by division into Cases.

Symbolizing a Situation to Find a Solution

Fallacies

A fallacy is an error in reasoning that results in an invalid argument.

Three common fallacies:

1) Using ambiguous premises, and treating them as if they were unambiguous.

2) Begging the question(assuming what is to be proved without have derived it from the premises).

3) jump to a conclusion(without adequate grounds)

4) converse error

5) inverse error

Contradictions and Valid Arguments

Summary of Rules of Inference

1.4 Application: Digital Logic Circuits

Black Boxes and Gates

Rules for a Combinational Circuit

Never combine two input wires.

A single input wire can be split partway and used as input

for two separate gates.

An output wire can be used as input

No output of a gate can eventually feed back into that gate.

multiple-input AND-gate

1.5 Application: Number Systems and Circuits for Application

Two‘s Complements and  the Computer Representation of Negative Integers

8-Bit Representation of a Number

Computer Addition With Negative Integers

Hexadecimal Notation