outline of discrete mathematics and its applications(1.1-1.3)

1.1 propositional logic (命题逻辑)

• proposition
  

  • propositional variables: use letters to denote
  • propositional calculus
  • compound propositions
  • true table(真值表)
  • logic operators are also called connectives
  • disjunction of p and q, denoted by p∪q called q or q. this or namely as an inclusive or and there is also exclusive or
    

  关于并集交集的真值表比较简单，注意exclusive or 的表达。

conditional statements

• conditional statements(also called implication)
• p -> q p is called the hypothesis(or antecedent or premise) q is called conclusion or consequence.
• p -> q 的真值表
• biconditional statement p<->q(also called bi-implications)
• p <->q 的真值表

truth table of compound propositions

precedence of logic operators

• 四个运算符运算从高到底如下 非 和 或 -> <->

logic and bit operations

1.2 applications of propositional logic

• translating english sentences
• system specifications
• boolean searches
• logic puzzles
• logic circuits(打不出来自己看书吧，很容易混)
• gates
  
  • NOT gate
  • OR gate
  • AND gate

1.3 propositional equivalences

• tautology(恒真命题,重言式)
• contradiction(矛盾式)
• contingency(翻译成一般命题？)
• logical equivalence: compound propositions that have the same truth values in all possible cases are called logically equivalent.
• logical equivalences 常用式子(基本运算重要啊！)
  
  • identity laws
  • domination laws
  • idempotent laws
  • double negation law
  • commutative laws
  • assosiative laws
  • distributive laws
  • de morgan’s laws
  • absorption laws
  • negation laws
• logical equivalences involving conditional statements
  
  • p -> q ≡ ￢p ∨ q(最重要的)
• logical equivalences involving biconditional statements
  
  • p <-> q ≡ (p∧q)∨(￢p∧￢q) (最重要的)