Understanding Boolean Logic: Propositions, Variables, and Logical Operators, Slides of Discrete Mathematics

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Boolean Logic

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Applications of Boolean logic

• Computer programs
• And computer addition
• Logic problems
• Sudoku

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• We use Boolean variables to refer to

propositions

• Usually are lower case letters starting with p (i.e. p, q, r, s , etc.)
• A Boolean variable can have one of two values true (T) or false (F)
• A proposition can be…
• A single variable: p
• An operation of multiple variables: p ∧( q ∨¬ r )

Boolean variables

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Introduction to Logical Operators

• About a dozen logical operators
  • Similar to algebraic operators + * - /
• In the following examples,
  • p = “Today is Friday”
  • q = “Today is my birthday”

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Logical operators: And

• An and operation is true if both operands are true
• Symbol: ∧
  • It’s like the ‘A’ in And
• In C++ and Java,

the operand is &&

• p ∧ q = “Today is Friday and

today is my birthday”

p q p ∧ q T T T T F F F T F F F F

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Logical operators: Or

• An or operation is true if either operands are true
• Symbol: ∨
• In C++ and Java,

the operand is ||

• p ∨ q = “Today is Friday or

today is my birthday (or possibly both)”

p q p ∨ q T T T T F T F T T F F F

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Inclusive Or versus Exclusive Or

• Do these sentences mean inclusive or

exclusive or?

• Experience with C++ or Java is required
• Lunch includes soup or salad
• To enter the country, you need a passport or a driver’s license
• Publish or perish

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Logical operators: Nand and Nor

• The negation of And and Or, respectively
• Symbols: | and ↓, respectively
  • Nand: p | q ≡ ¬( p ∧ q )
  • Nor: p ↓ q ≡ ¬( p ∨ q )

p q p ∧ q p ∨ q p | q p ↓ q T T T T F F T F F T T F F T F T T F F F F F T T

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Logical operators: Conditional 2

• Let p = “I am elected” and q = “I will lower taxes”
• I state: p → q = “If I

am elected, then I will lower taxes”

• Consider all

possibilities

• Note that if p is false, then

the conditional is true regardless of whether q is true or false

p q p → q T T T T F F F T T F F T

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Logical operators: Conditional 3

Conditional Inverse Converse Contra- positive p q ¬ p ¬ q p → q ¬ p →¬ q q → p ¬ q →¬ p

T T F F T T T T

T F F T F T T F

F T T F T F F T

F F T T T T T T

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Logical operators: Bi-conditional 1

• A bi-conditional means “ p if and only if q ”
• Symbol: ↔
• Alternatively, it means “(if p then q ) and (if q then p )”
• p ↔ q ≡ p → q ∧ q →p
• Note that a bi-conditional has the opposite truth values of the exclusive or

p q p ↔ q T T T T F F F T F F F T

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Logical operators: Bi-conditional 2

• Let p = “You take this class” and q = “You get a grade”
• Then p ↔ q means “You take this class if and only if you get a grade”
• Alternatively, it means “If you take this class, then you get a grade and if you get a grade then you take (took) this class”

p q p ↔ q T T T T F F F T F F F T

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Precedence of operators

• Just as in algebra, operators have

precedence

• 4+32 = 4+(32), not (4+3)*
• Precedence order (from highest to lowest):

• The first three are the most important
• This means that p ∨ q ∧ ¬ r → s ↔ t

yields: ( p ∨ ( q ∧ (¬ r ))) ↔ ( s → t )

• Not is always performed before any other

operation

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Translating English Sentences

• Problem:
  • p = “It is below freezing”
  • q = “It is snowing”
• It is below freezing and it is snowing
• It is below freezing but not snowing
• It is not below freezing and it is not snowing
• It is either snowing or below freezing (or both)
• If it is below freezing, it is also snowing
• It is either below freezing or it is snowing, but it is not snowing if it is below freezing
• That it is below freezing is necessary and sufficient for it to be snowing

p ∧ q p ∧¬ q ¬ p ∧¬ q p ∨ q p → q ( p ∨ q )∧( p →¬ q )

p ↔ q