Equivalent Statements, 3.4

• Two statements are equivalent, symbolized by ⇔, if both statements have exactly the same final column in truth tables.
• There are two famous equations in logic (also in set theory) called De Morgan's Laws.
  • ~(p ∧ q) ⇔ ~p ∨ ~q
  • ~(p ∨ q) ⇔ ~p ∧ ~q
• Let's demonstrate that one of these two statements is true.
• De Morgan's can be applied to more complex statements.
  • It is not true that the sun is blue or the sky is not green.
  • p: the sun is blue
  • q: the sky is green
  • Original statement ~(p ∨ ~q)
  • Apply DE Morgans
    • Change the and to or
    • Negate the simple statements
  • ~p ∧ q
  • The sun is not blue and the sky is green
  • ~(~p ∧ ~q) becomes p ∨ q
  • p ∧ ~q becomes ~(~p ∨ q)
  • In general, when translating DE Morgan's statements I would
    • Represent the statement in symbolic form.
    • Apply DE Morgan's
    • Translate back to English form.
• Show that p → q ⇔ ~p ∨ q