module 2

• • Logic the foundation on which the discipline is built
• Mathematical Statement is a statement that can be assigned a truth value and classified as true or false, but not both.
• • Mathematical Statements are Declarative Sentences.
  • We shall not consider declarative statements whose truth value is not clear or a matter of opinion as mathematical statements.
  • Questions, exclamations, and imperatives are not considered as mathematical statements as well, since these sentences do not have a truth value.
• Operations on Statements: The connects

  • Conjunction: p and q - p ∧ q
  • Disjunction: p or q - p ∨ q
  • Conditional: If p then q - p → q
  • Biconditional: p if and only if q - p ↔ q
  • Negation: not p - ∼ p
• • All operations are binary (involves two statement) except the negation.
  • In the conditional statement p → q, p is called the Premise while q is the Conclusion.
  • In the biconditional statement p ↔ q, means (p → q) ∧ (q → p).
• For the operations on mathematical statements to be well-defined operations, the result must also be a mathematical statement that is, it must have a truth value.
• The conjunction p ∧ q is true if both p and q are true. Otherwise, it is false.
• The disjunction p ∨ q is true if at least one statement (p, q or both) is true. It is false only if both statements are false.
• The conditional p → q is false only when the premise p is true and the conclusion q is false. Otherwise, it is true.
• The biconditional p ↔ q is true if p and q have the same truth value, that is, either p and q are both true or false.
• The negation ∼ p is true if p is false. If p is true then ∼ p is false.
• In cases of operations on compound statements, we use delimiters, ( ), { }, [ ] to group statements together.
• Truth tables can be used to represent the truth values of the compound statements we have discussed. We consider all possible cases for p and q (True (T) or False (F)).
• • Conjunction: True only when both p and q are true.
• Disjunction: False only when both p and q are false.
• Conditional: False only when the premise is true and conclusion is false.
• Biconditional: True only when p and q have the same truth value
• Implication: If a statement p (materially) implies statement q, we denote this by p ⇒ q (read as p implies q)
• Equivalence exists if two compound statements have the same truth value. We denote the equivalence of the two statements p and q by p ⇔ q.
• To negate simple statements, we usually add the word ”not”, as appropriate.
• If there are two statements, there are 2^2 = 4 rows in the table showing all possible cases.
• If there are three statements, there are 2^3 = 8 rows in the table showing all possible cases.
• If there are n statements, there are 2^n cases
• In general, for statements with adjectives, replacing the adjective with its antonym or another adjective is not equivalent to negating the statement
• To negate a statement means to determine the opposite of a statement.
• Negation of Compound Statements

  To negate a conjunction or disjunction, we use the following equivalences:

  • ∼ (p ∧ q) ⇔∼ p∨ ∼ q
  • ∼ (p ∨ q) ⇔∼ p∧ ∼ q
• Negation of Statements with Quantifiers We can also negate statements with quantifiers all, some, none.
• Negation:

  • All roses are red.

  Negation: Not all roses are red.

  Negation: Some roses are not red.

  • Some roses are red.

  Means there are roses which are red. The negation is: there are no roses which are red, that is Negation: No roses are red.

  
  

  • No roses are red.

  Negation: Some roses are red. DPSM (UP Visayas) Logic and Reasoning
• Equivalent forms for the Conditional

  The conditional p → q or ”If p then q” is equivalent to the following statements:

  • q if p
  • p only if q
  • p is sufficient for q
  • q is necessary for p
  • All p are q
  • Either not p or q
• Given the conditional p → q, then the

  • Contrapositive is ∼ q →∼ p

  • Converse is q → p

  • Inverse is ∼ p →∼ q
• An Euler Diagram represents statements, the way Venn Diagram represents sets.
• Valid Argument

  An Argument consists of premises, say p1, p2, . . . , pn, and a conclusion q. Consider the conjunction

  p1 ∧ p2 ∧ . . . ∧ pn = p

  The argument p → q is valid if the premises are assumed to be true, then the conclusion must hold true. That is, the statement p → q is an implication
• • The argument is valid if the conclusion is satisfied by the Euler diagram representing all the premises.
  • The premises are assumed to be true although statements may not be true in the strict sense.
• Invalid Argument

  To show an argument is invalid, it suffices to exhibit an Euler diagram satisfying all the premises but not the given conclusion.
• • One may also show that an argument is invalid by exhibiting two different diagrams representing the premises.
  • If an argument is valid, there should only be one possible conclusion
• • The argument may be valid even if the premises are not universally true. That is, the argument is valid but not “sound”.
  • We can have premises that may be meaningless, but the conclusion (which is also meaningless) can still follow logically if the premises are assumed to be true.
• modus ponens

  • All dogs are hairy. Cotton is a dog. Therefore, Cotton is hairy.
  • [(p → q) ∧ p] → q
• Modus tollens

  • All dogs are hairy. My pet Donut is not hairy. Therefore, Donut is not a dog.
  • [(p → q)∧ ∼ q] →∼ p
• syllogism

  • All cats are mammals. All mammals are animals. Therefore, all cats are animals
  • [(p → q) ∧ (q → r)] → (p → r)
• fallacy of the converse

  • All dogs are hairy. My pet Cotton is hairy. Therefore, Cotton is a dog.
  • [(p → q) ∧ q] → p