Section 3.3 Conditional and Biconditional Statements.

• Conditional statements.
  • A conditional statement involves if and then.
  • If today is Monday then you are in class.
  • The symbol is →
  • p: Today is Monday
  • q: You are in class
  • p → q
• Translating conditional statements.
  • If you go to the game then you will not see a movie.
  • If you do not eat lunch then you will not take a nap.
• The truth table for conditional statements.
  • Think If you get an A then I will buy you ice cream.
    • Did I say what I would do if you don't get an A?
    • So no matter what happens if you don't get an A, the statement is true.
    • If you get an A and I get you ice cream is the statement true?
    • How about if you get an A but I don't get you ice cream? That is the problem.

    • p	q	p → q
      T	T	T
      T	F	F
      F	T	T
      F	F	T

  • Note, this is NOT commutative
    • p → q is not the same as q → p.
    • IN THIS ONE THE ORDER MATTERS
  • Translate to symbolic form and build a truth table for the following
    • If it is not cold then we can go for a walk.
    • It is not true that if the food is cold then we will not eat.
    • If the sky is cloudy and the snow is falling then we will not have a picnic.
  • Biconditional
    • A biconditional is a statement involving if and only if.
    • the symbol is ↔
    • You will pass the class if and only if your average is above a 60%
    • p: you will pass the class
    • q: your average is above a 60%
    • p ↔ q
  • The truth table

    • p	q	p ↔ q
      T	T	T
      T	F	F
      F	T	F
      F	F	T

    • If the values match, the statement is true.
• A self contradiction is a compound statement that is always false.
  • Build the truth table for ~p ∧ (q ↔ ~q)
• A tautology is a compound statement that is always true
  • Build the truth table for (p ∧ ~q) ↔ ~(p → q)
• An implication is a conditional statement that is a tautology.
  • Build the truth table for p → (q ∨ q)