Statements and Logical Connectives

• A statement is a sentence which can be judged to be true or false.
• A simple statement conveys only one idea.
• A compound statement is a combination of simple statements.
• Write some simple statements
• Represent the statements symbolically.
• Compound statements is one or more simple statements containing any of the following :
  • not
  • and
  • or
  • if then
  • if and only if.
• Negation or NOT
  • Unary operator.
  • ~p
  • Truth table.
  • Careful if ALL, SOME, NONE or equivalents are involved.
  • The negation of a statement should have the opposite truth value.
  • All lakes have fresh water.
    • No lakes have fresh water
    • Some lakes have fresh water.
  • All are becomes some are not
  • None are becomes some are
  • We will only let letters represent simple positive statements.
• Conjunction or AND
  • ∧
  • Symbolic representation
  • truth table.
• Disjunction or OR
  • ∨
  • Symbolic representation.
  • Truth table.
• If a statement contains multiple connectives, the comma shows us where to put parenthesis.
• Negation only negates the symbol which immediately follows, to negate a compound statement we must use ()
• It is not true that <==> ~()
• Conditional or IF-THEN
  • If it rains then we will get wet.
  • →
  • Truth table.
• Biconditional or IF-AND-ONLY-IF
  • You will pass if and only if your grade is above 60%
  • ↔
• A compound statement that is always true is a tautology.
• A compound statement that is always false is a self-contradiction
• A conditional statement that is a tautology is an implication.