Equivalent Statements

• Two statements are equivalent symbolized ⇔ if they have exactly the same truth values in the final columns of the truth table.
• This is sometimes called logically equivalent
• To show this
  • Build a truth table for each statement
  • Compare the final column.
• De Morgan's Laws
  • ~(p ∧ q) ⇔ ~p ∨ ~q
  • ~(p ∨ q) ⇔ ~p ∧ ~q
  • Show these to be true.
• Practical De Morgan's
  • Negate: ~p ∨ q ⇔ ~(~~p ∧ ~q) ⇔ ~(p ∧ q)
• Applying De Morgan's Laws
  • Write the statements in symbolic form.
  • Negate the statement
  • Apply De Morgan's
  • Rewrite as statements.
  • Negate: It is false that a rabbit's foot is lucky and an elephant foot is not lucky.
  • Negate: A cow is not a reptile or a duck is a plant.
• The following is true
  • p → q ⇒ p ∨ q
  • Show this
  • Thus with De Morgan's ~(p → q) ⇒ p ∧ ~q
  • Negate: If a duck is a plant then an elephant foot is not lucky.
• They give a number of equivalent statements. I'm not too worried about these.