Statements and Logical Connectives

• A statement is a sentence which can be judged true or false.
  • Today is Tuesday
  • Rain is falling
  • Today is Tuesday and rain is falling.
• The first two are examples of simple statements.
• The third is an example of a compound statement.
• We can represent a simple statement as a variable.
  • Normally we choose p,q,r, ....
  • p: Today is Tuesday.
  • q: Rain is falling.
• We will always represent statements in the positive.
• We have a number of words we use to modify simple statements.
• Not or negation
  • Today is not Tuesday
  • It is not Raining.
  • This is really the idea of a compliment in set notation.
  • We use the symbol ~
  • p: Today is Tuesday, ~p : Today is not Tuesday.
  • For any statement s, either s is true and ~s is false, or s is false and ~s is true.
• Quantifiers
  • Be careful with ALL, NONE, SOME
  • All frogs are green.
  • This implies that every single frog is green.
  • The negation is not NO frogs are green.
  • The first statement is false if all but one frog is green.
  • This statement is false in that case too.
  • The correct negation is Some frogs are not green.
  • Some are maps to NONE are
  • All are maps to some are not.
• Truth tables is a device to evaluate the logic of a compound statement.
• The truth table for not is
• A conjunction involves the word and, along with two statements.
  • Today is Tuesday and rain is falling.
  • we use the symbol ∧
  • p ∧ q;
  • The basic truth table.
• A disjunction involves the work or, along with two statements.
  • Today is Tuesday or rain is falling.
  • we use the symbol ∨
  • p ∨ q.
  • In this case, we use the inclusive or.
  • But there is an exclusive or too.
• A conditional is an if then statement
  • If today is Tuesday then rain is falling.
  • we use the symbol →
  • p → q
  • If you get an A, then I will buy you a car.
  • Truth table.
• A biconditional statement is an if and only if statement.
  • Today is Tuesday if and only if rain is falling
  • p ↔q
  • Truth table.
• The placement of a comma is used to determine where parenthesis are placed.
  • Today is Tuesday and rain is falling, or class is fun.
  • Today is Tuesday, and rain is falling or class is fun.