Geometry Unit 1: Logic Concepts

Logic

• Introduction to Logic

  • Distinction between mathematical and nonmathematical sentences
  • Open sentences contain variables; closed sentences do not

• Negation

  • Symbol: ~

• Conjunctions (And)

  • Symbol: ∧
  • Truth table:
    • T, T → T
    • T, F → F
    • F, T → F
    • F, F → F

• Disjunctions (Or)

  • Symbol: ∨
  • Truth table:
    • T, T → T
    • T, F → T
    • F, T → T
    • F, F → F

• Conditionals (If..., then...)

  • Symbol: →
  • Truth table:
    • T, T → T
    • T, F → F
    • F, T → T
    • F, F → T

• Biconditionals (...if and only if...)

  • Symbol: ↔
  • Truth table:
    • T, T → T
    • T, F → F
    • F, T → F
    • F, F → T

• Tautology

  • A compound statement that is always true

• Logically Equivalent Statements

  • Statements that yield the same truth value in all possible scenarios

• De Morgan's Laws

  • ~(p ∧ q) is equivalent to ~p ∨ ~q
  • ~(p ∨ q) is equivalent to ~p ∧ ~q

• Conditional Statements and Related Forms

  • Conditional: p → q
  • Converse: q → p
  • Inverse: ~p → ~q
  • Contrapositive: ~q → ~p
  • Conditionals and contrapositives are logically equivalent: (p → q) ↔ (~q → ~p)

• Law of Contrapositives

  • From p → q concludes ~q → ~p

• Law of Modus Ponens (Detachment)

  • Given p → q and p, conclude q

• Law of Modus Tollens

  • Given p → q and ~q, conclude ~p

• Law of Disjunctive Inference

  • Given p ∨ q and ~p, conclude q
  • Given p ∨ q and ~q, conclude p

• Law of Conjunction

  • From p and q conclude p ∧ q

• Law of Simplification

  • From p ∧ q, conclude p
  • From p ∧ q, conclude q

• Law of Disjunctive Addition

  • From p, conclude p ∨ q

• Chain Rule (Law of Syllogism)

  • From p → q and q → r, conclude p → r