Lesson 1: Elementary Logic

• This lesson deals with the relationship between the natural language and the mathematical language using symbols and variables.

  Elementary Logic
• It is a statement that has a truth value.

  Proposition
• Two possible truth values associated with propositions:

  True (T) & False (F)
• It is a single statement that does not contain other statements as parts.

  Simple Statement
• It contains two or more statements.

  Compound Statement
• It combines simple statements into compound statement.

  Logical Connective (Propositional Operator)
• Connective: Not

  Type of Statement: Negation

  Symbol: ~

  Symbolic Form: ~p

  Read: Not p (p is false)
• Connective: And/But

  Type of Statement: Conjunction

  Symbol: ^

  Symbolic Form: p ^ q

  Read: p and q (both p and q are true)
• Connective: Or

  Type of Statement: Disjunction

  Symbol: V

  Symbolic Form: p V q

  Read: p or q (either 𝑝 is true or 𝑞 is true or both 𝑝 and 𝑞 are true)
• Connective: Implies

  Type of Statement: Conditional

  Symbol: ->

  Symbolic Form: p -> q

  Read: If p then q
• Connective: If and only if

  Type of Statement: Biconditional

  Symbol: <->

  Symbolic Form: p <-> q

  Read: p if and only if q
• A logical statement may either be true or false. If the statement is true, then the truth value of that statement is true and is denoted by T. If it is false, then its truth value is false and is denoted by F.

  Truth Table
• Two mathematical statements are logically equivalent if they have the same truth values in all possible cases.
• Converse: of p -> q is q -> p.

  Inverse: of p -> q is ~p -> ~q.

  Contrapositive: of p -> q is ~q -> ~p.