Truth Tables and Operations in Boolean Algebra

• A truth table allows us to evaluate all possible values for a Boolean expression.
• We will first use truth tables to show the value of the operations (think addition table or multiplication table)
• We will then use truth tables to evaluate Boolean expressions.
• There are 5 you need to memorize.
  • You should put them on your cheat sheet
  • But that will be slow to use.
  • Fortunately they are easy.
• A truth table shows input on the left hand side, and output on the right hand side.
• Negation
  • The truth table for negation (not)

    p	~p
    T	F
    F	T

  • Negation flips the truth of the statement.
    • Negate a true statement it becomes false
    • Negate a false statement it becomes true.
• Conjunction (and)
  • And connects two statements.
  • The result is true only if both statements are true.
  • The truth table has 2 inputs.
  • And four rows.

  • p	q	p∧q
    T	T	T
    T	F	F
    F	T	F
    F	F	F

• Disjunction (or)
  • Or connects two statements.
  • The result is true if one, or the other or both are true

  • p	q	p∨q
    T	T	T
    T	F	T
    F	T	T
    F	F	F

• Conditional (if then)
  • Conditional connects two statements.
  • This one is a little different

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

  • Think: If you get an A, I will buy you an ice cream cone.
• Biconditionl (if and only if)
  • Or connects two statements.
  • The result is true if the inputs are the same.

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

  • Think I will buy you an ice cream cone if and only if you get an A.
• Constructing A truth table
  1. Build the input
     • For i variables, there will be 2ⁱ rows.
       • 1 variable - 2 rows
       • 2 variables - 4 rows
       • 3 variables - 8 rows.
       • I won't go higher than this.
     • Start with T, work to false.
       • T - F
       • TT - TF - FT - FF
       • TTTT - TTF - TFT - TFF - FTT - FTF - FFT - FFF
       • Notice the last column alternates: T-F-T-F
       • The second alternates by 2 : Tx - Tx - Fx - Fx
       • The third alternates by 4 : Txx - Txx - Txx - Txx - Fxx - Fxx - Fxx - Fxx
  2. Build the output
     • Two methods, build by order of operations
     • Build under the equation.
• Do some exercises P 115
  • 5-26
  • 27-52
• Do some exercises P 124
  • 7-30
• A self contradiction is a compound statement that is always false
  • p ∧ ~p
• A tautology is a compound statement that is always true
  • p ∨ ~p
• A implication is a conditional statement that is a tautology
  • ((p ∧ q) ∧ p) → q
• Do some exercises P 125 31-42