Equivalent Statements

• Two statements are equivalent, written ⇔, if both statements have exactly the same values in the "answer" columns of the truth tables.
• There are De Morgan's laws (stated by Augustus De Morgan.
  • These apply to sets as well, but I skipped them in that chapter.
  • ~(p ∧ q) ⇔ ~p ∨ ~q
  • ~(p ∨ q) ⇔ ~p ∧ ~q
• Show that these two are equivalent.
• Using De Morgan's
  • ~(p ∧ ~q) ⇔ ~p ∨ q
  • ~(~p ∧ q) ⇔ p ∨ ~q
  • ~(~p ∧ ~q) ⇔ p ∨ q
  • Negate the statement : Today is Wednesday or snow is not falling:

    p: Today is Wednesday.
    q: Snow is falling.
    S: p ∨ ~q
    ~S: ~(p ∨ ~q) ⇔ ~p ∧ q
    Today is not Wednesday and Snow is falling.
    	

• Some additional equivalent statements
  • p → q ⇔ ~p ∨ q
  • ~(p → q) ⇔ p ∧ ~q, we can easily show this.
  • p → q ⇔ ~q → ~p, this is called contrapositive
  • If today is Thursday then we are in class. ⇔ If we are not in class then today is not Thursday.
• Do problems page 136 1-48