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)