離散數學數理邏輯部分部分稿件loading

2019-11-06 08:50:30

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數理邏輯

chapter 2 Logic

2.1 PRopositions(命題) and logical Operations

part 1 propositions

a statement or proposition is a declarative sentence（陳述句） that is either true or false, but not both.

special example:

the temperature on the surface of the planet venus is 900.F the sun will come out tomorrow.

they are propositions.

part 2 logical connectives and compound(復合) statement.

in logic , the letters p,q,r …denote propositional variables:that is , variable that can be replaced with statements,statements or propositional variables can be combined by logical connectives to obtain compound statements, . and :tthe truth value of a compound statements depends only on the truth values of the statements being combined and on the types of connectives being used.

negation:

the negation of p is the statement not p , denoted by ~p . it follows that if p is true, then ~p is flase, if p is false , then ~p is true. truth table:table giving the truth values of a compound statement in terms of its compound parts , is called truth table. note: not is not a connective , sice it doesn’t join two statements. ~p is not a compound statement. however , ~ is a unary operation for the collection of statements and ~p is a statement if p is.

conjunction

definition:if p and q are statements , the conjunction of p and q is the compound statement “p and q , denoted as the p ∧ $/land$ q”. and is a binary operation on the set of statements , the compound statement p ∧ $/land$ q is true when both p and q are true, otherwise it is false. we may join two totally unreleated statements by the connective and.

disjunction

definition:if p and q are statements , the disjunction of p and q is the compound statement “p or q”, denoted by p ∨ $/vee$ q. the compound statement is true if p or q is true, it is false when both p and q are false.

truth table

constructed: 1. step1:the first n columns of the table are lableed by the component propositional variables , further columns are included for all intermediate combinations of the variables,culmuinating in a column for the full statement. 2. under each of the first headings , we list the 2ⁿ possible n-tuples of truth values for the n compound statements. 3.for each of the remaining columns, we compute , in sequence, the remaining truth values.

qualifiers

an element of {x|P(x)} is an object t for which the statement P(t) is true,such a statement P(x) is called a predicate,P(x) is also called a propositional function , because each choice of x produce a proposition P(x) thatt is eitherr true or flase, another use of predicates is in programming, like if(P(x)) , the predicates P(x) are called the guards for the block of programming code, often the guard for a block is a conjunction or disjunction. the universal quantification of a predicate ? $/forall$ ,means that for all values of x P(x) is true.denoted by ? $/forall$ x P(x) the existial quantification:their exist a value of x for which P(x) is true,denoted by ? $/exists$ .

2.2Conditional Statements

if p and q are statements , the compound statement “if p then q”, denoted p?q $p /Rightarrow q$ , is called a conditional statement , or implication(蘊含)。 the statement p is called the antecedent(前件) or hypothesis(前提)。and the statement q is called the consequent or conclusion. the connective if…then is denoted by the symbol ? $/Rightarrow$ p?q $p /Rightarrow q$ the converse q?p $q /Rightarrow p$ and the contrapositive q? p $~q /Rightarrow ~p$