Applied Discrete Mathematics

1
More About Logic

• Logic is a system based on propositions.
• A proposition is a statement that is either true
  or false (not both).
• We say that the truth value of a proposition is
  either true (T) or false (F).
• Corresponds to 1 and 0 in digital circuits

2
Logical Operators (Connectives)

• We will examine the following logical operators
• Negation (NOT)
• Conjunction (AND)
• Disjunction (OR)
• Exclusive or (XOR)
• Implication (if then)
• Biconditional (if and only if)
• Truth tables can be used to show how these
  operators can combine propositions to compound
  propositions.

3
Negation (NOT)

• Unary Operator, Symbol ?

4
Conjunction (AND)

• Binary Operator, Symbol ?

5
Disjunction (OR)

• Binary Operator, Symbol ?

6
Exclusive Or (XOR)

• Binary Operator, Symbol ?

7
Implication (if - then)

• Binary Operator, Symbol ?

8
Biconditional (if and only if)

• Binary Operator, Symbol ?

9
Statements and Operators

• Statements and operators can be combined in any
  way to form new statements.

10
Statements and Operations

• Statements and operators can be combined in any
  way to form new statements.

11
Equivalent Statements

• The statements ?(P?Q) and (?P)?(?Q) are logically
  equivalent, because ?(P?Q)?(?P)?(?Q) is always
  true.

12
Converse, Contrapositive, Inverse
Given P?Q . . .
Converse Q?P
Contrapositive ?Q? ?P
Inverse ?P? ?Q
13
Tautologies and Contradictions

• A tautology is a compound statement that is
  always true.
• Examples
• R?(?R)
• ?(P?Q)?(?P)?(?Q)
• If S?T is a tautology, we write S?T.
• If S?T is a tautology, we write S?T This symbol
  is also used for logical equivalence.

14
Tautologies and Contradictions

• A contradiction is a compound statement that is
  always false.
• Examples
• R?(?R)
• ?(?(P?Q)?(?P)?(?Q))
• The negation of any tautology is a contra-
• diction, and the negation of any contradiction is
  
• a tautology.

15
Exercises

• We already know the following tautologies known
  as
• De Morgans Laws
• ?(P?Q) ? (?P)?(?Q)
• ?(P?Q) ? (?P)?(?Q) using a truth table.
• Classroom Exercise Show the following using a
  truth table
• (P?Q)?R ? P?(Q?R) (Associative Laws)
• (P?Q)?R ? P?(Q?R)
• Table 6 in Section 1.2 shows many useful laws.
• Exercises 1 and 7 in Section 1.2 may help you
  get used to propositions and operators.

16
Classroom Exercises

• Using truth tables, show the rest of the logical
  equivalences in tables 5, 6, and 7

17
Homework

• Section 1.2 5, 9, 22, 23
• Read 1.3